Dirichlet series
| L(s) = 1 | − 3-s + 3·5-s − 7·9-s + 9·11-s + 7·13-s − 3·15-s + 3·17-s + 7·19-s − 23-s − 17·25-s + 12·27-s + 18·29-s + 30·31-s − 9·33-s + 23·37-s − 7·39-s − 15·41-s + 12·43-s − 21·45-s − 16·47-s − 3·51-s + 33·53-s + 27·55-s − 7·57-s − 10·59-s + 61-s + 21·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 7/3·9-s + 2.71·11-s + 1.94·13-s − 0.774·15-s + 0.727·17-s + 1.60·19-s − 0.208·23-s − 3.39·25-s + 2.30·27-s + 3.34·29-s + 5.38·31-s − 1.56·33-s + 3.78·37-s − 1.12·39-s − 2.34·41-s + 1.82·43-s − 3.13·45-s − 2.33·47-s − 0.420·51-s + 4.53·53-s + 3.64·55-s − 0.927·57-s − 1.30·59-s + 0.128·61-s + 2.60·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{30} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{30} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{30} \cdot 7^{30} \cdot 41^{15}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28752\times 10^{27}\) |
| Root analytic conductor: | \(8.01047\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((30,\ 2^{30} \cdot 7^{30} \cdot 41^{15} ,\ ( \ : [1/2]^{15} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(880.1245006\) |
| \(L(\frac12)\) | \(\approx\) | \(880.1245006\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 41 | \( ( 1 + T )^{15} \) | |
| good | 3 | \( 1 + T + 8 T^{2} + p T^{3} + 13 p T^{4} + 31 T^{5} + 179 T^{6} + 71 p T^{7} + 202 p T^{8} + 916 T^{9} + 1919 T^{10} + 1232 p T^{11} + 6947 T^{12} + 4168 p T^{13} + 7016 p T^{14} + 11713 p T^{15} + 7016 p^{2} T^{16} + 4168 p^{3} T^{17} + 6947 p^{3} T^{18} + 1232 p^{5} T^{19} + 1919 p^{5} T^{20} + 916 p^{6} T^{21} + 202 p^{8} T^{22} + 71 p^{9} T^{23} + 179 p^{9} T^{24} + 31 p^{10} T^{25} + 13 p^{12} T^{26} + p^{13} T^{27} + 8 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) |
| 5 | \( 1 - 3 T + 26 T^{2} - 3 p^{2} T^{3} + 351 T^{4} - 983 T^{5} + 3481 T^{6} - 1903 p T^{7} + 5812 p T^{8} - 75678 T^{9} + 209221 T^{10} - 513824 T^{11} + 1309803 T^{12} - 3067058 T^{13} + 7286756 T^{14} - 16275117 T^{15} + 7286756 p T^{16} - 3067058 p^{2} T^{17} + 1309803 p^{3} T^{18} - 513824 p^{4} T^{19} + 209221 p^{5} T^{20} - 75678 p^{6} T^{21} + 5812 p^{8} T^{22} - 1903 p^{9} T^{23} + 3481 p^{9} T^{24} - 983 p^{10} T^{25} + 351 p^{11} T^{26} - 3 p^{14} T^{27} + 26 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 11 | \( 1 - 9 T + 89 T^{2} - 498 T^{3} + 2998 T^{4} - 12910 T^{5} + 62896 T^{6} - 244295 T^{7} + 1118943 T^{8} - 4242745 T^{9} + 18355959 T^{10} - 5944524 p T^{11} + 257372053 T^{12} - 841875891 T^{13} + 280213963 p T^{14} - 9581848759 T^{15} + 280213963 p^{2} T^{16} - 841875891 p^{2} T^{17} + 257372053 p^{3} T^{18} - 5944524 p^{5} T^{19} + 18355959 p^{5} T^{20} - 4242745 p^{6} T^{21} + 1118943 p^{7} T^{22} - 244295 p^{8} T^{23} + 62896 p^{9} T^{24} - 12910 p^{10} T^{25} + 2998 p^{11} T^{26} - 498 p^{12} T^{27} + 89 p^{13} T^{28} - 9 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 - 7 T + 76 T^{2} - 375 T^{3} + 2621 T^{4} - 10793 T^{5} + 61717 T^{6} - 220271 T^{7} + 1122185 T^{8} - 3552915 T^{9} + 17109259 T^{10} - 48621973 T^{11} + 230086024 T^{12} - 601963629 T^{13} + 2936644665 T^{14} - 7551218442 T^{15} + 2936644665 p T^{16} - 601963629 p^{2} T^{17} + 230086024 p^{3} T^{18} - 48621973 p^{4} T^{19} + 17109259 p^{5} T^{20} - 3552915 p^{6} T^{21} + 1122185 p^{7} T^{22} - 220271 p^{8} T^{23} + 61717 p^{9} T^{24} - 10793 p^{10} T^{25} + 2621 p^{11} T^{26} - 375 p^{12} T^{27} + 76 p^{13} T^{28} - 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 - 3 T + 50 T^{2} - 128 T^{3} + 1899 T^{4} - 4363 T^{5} + 49157 T^{6} - 88690 T^{7} + 1034344 T^{8} - 1427757 T^{9} + 18105190 T^{10} - 15514456 T^{11} + 284744193 T^{12} - 98181573 T^{13} + 4458511390 T^{14} - 907579244 T^{15} + 4458511390 p T^{16} - 98181573 p^{2} T^{17} + 284744193 p^{3} T^{18} - 15514456 p^{4} T^{19} + 18105190 p^{5} T^{20} - 1427757 p^{6} T^{21} + 1034344 p^{7} T^{22} - 88690 p^{8} T^{23} + 49157 p^{9} T^{24} - 4363 p^{10} T^{25} + 1899 p^{11} T^{26} - 128 p^{12} T^{27} + 50 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 - 7 T + 118 T^{2} - 440 T^{3} + 5385 T^{4} - 8099 T^{5} + 160785 T^{6} + 56883 T^{7} + 4342870 T^{8} + 8044488 T^{9} + 105632963 T^{10} + 341081490 T^{11} + 2256115895 T^{12} + 9680538834 T^{13} + 45674046668 T^{14} + 205914593365 T^{15} + 45674046668 p T^{16} + 9680538834 p^{2} T^{17} + 2256115895 p^{3} T^{18} + 341081490 p^{4} T^{19} + 105632963 p^{5} T^{20} + 8044488 p^{6} T^{21} + 4342870 p^{7} T^{22} + 56883 p^{8} T^{23} + 160785 p^{9} T^{24} - 8099 p^{10} T^{25} + 5385 p^{11} T^{26} - 440 p^{12} T^{27} + 118 p^{13} T^{28} - 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 + T + 100 T^{2} - 240 T^{3} + 5823 T^{4} - 23139 T^{5} + 307392 T^{6} - 1276767 T^{7} + 12804893 T^{8} - 57452087 T^{9} + 449235292 T^{10} - 2003430672 T^{11} + 13871296909 T^{12} - 58567273319 T^{13} + 364132115290 T^{14} - 1474508609730 T^{15} + 364132115290 p T^{16} - 58567273319 p^{2} T^{17} + 13871296909 p^{3} T^{18} - 2003430672 p^{4} T^{19} + 449235292 p^{5} T^{20} - 57452087 p^{6} T^{21} + 12804893 p^{7} T^{22} - 1276767 p^{8} T^{23} + 307392 p^{9} T^{24} - 23139 p^{10} T^{25} + 5823 p^{11} T^{26} - 240 p^{12} T^{27} + 100 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 - 18 T + 351 T^{2} - 3938 T^{3} + 45998 T^{4} - 391789 T^{5} + 3532456 T^{6} - 25324251 T^{7} + 6769578 p T^{8} - 1257094469 T^{9} + 303500612 p T^{10} - 51641736178 T^{11} + 332842157595 T^{12} - 1810862813908 T^{13} + 10895718454521 T^{14} - 1919708619818 p T^{15} + 10895718454521 p T^{16} - 1810862813908 p^{2} T^{17} + 332842157595 p^{3} T^{18} - 51641736178 p^{4} T^{19} + 303500612 p^{6} T^{20} - 1257094469 p^{6} T^{21} + 6769578 p^{8} T^{22} - 25324251 p^{8} T^{23} + 3532456 p^{9} T^{24} - 391789 p^{10} T^{25} + 45998 p^{11} T^{26} - 3938 p^{12} T^{27} + 351 p^{13} T^{28} - 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 - 30 T + 729 T^{2} - 12702 T^{3} + 192287 T^{4} - 2469077 T^{5} + 28541588 T^{6} - 295094480 T^{7} + 2801338765 T^{8} - 24357725643 T^{9} + 196723756650 T^{10} - 47527253130 p T^{11} + 10319381498616 T^{12} - 67463739856866 T^{13} + 413955859752302 T^{14} - 2377693087445112 T^{15} + 413955859752302 p T^{16} - 67463739856866 p^{2} T^{17} + 10319381498616 p^{3} T^{18} - 47527253130 p^{5} T^{19} + 196723756650 p^{5} T^{20} - 24357725643 p^{6} T^{21} + 2801338765 p^{7} T^{22} - 295094480 p^{8} T^{23} + 28541588 p^{9} T^{24} - 2469077 p^{10} T^{25} + 192287 p^{11} T^{26} - 12702 p^{12} T^{27} + 729 p^{13} T^{28} - 30 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 - 23 T + 400 T^{2} - 5145 T^{3} + 58100 T^{4} - 567646 T^{5} + 5181174 T^{6} - 43727378 T^{7} + 355029388 T^{8} - 2734178148 T^{9} + 20402693048 T^{10} - 144984533863 T^{11} + 997358220188 T^{12} - 6561885310287 T^{13} + 41976916551657 T^{14} - 258407011673340 T^{15} + 41976916551657 p T^{16} - 6561885310287 p^{2} T^{17} + 997358220188 p^{3} T^{18} - 144984533863 p^{4} T^{19} + 20402693048 p^{5} T^{20} - 2734178148 p^{6} T^{21} + 355029388 p^{7} T^{22} - 43727378 p^{8} T^{23} + 5181174 p^{9} T^{24} - 567646 p^{10} T^{25} + 58100 p^{11} T^{26} - 5145 p^{12} T^{27} + 400 p^{13} T^{28} - 23 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 - 12 T + 269 T^{2} - 2316 T^{3} + 34068 T^{4} - 253607 T^{5} + 3153596 T^{6} - 21954583 T^{7} + 240692362 T^{8} - 1568196355 T^{9} + 358413070 p T^{10} - 94482346296 T^{11} + 19861613581 p T^{12} - 4960571657906 T^{13} + 41710893999939 T^{14} - 228266308319210 T^{15} + 41710893999939 p T^{16} - 4960571657906 p^{2} T^{17} + 19861613581 p^{4} T^{18} - 94482346296 p^{4} T^{19} + 358413070 p^{6} T^{20} - 1568196355 p^{6} T^{21} + 240692362 p^{7} T^{22} - 21954583 p^{8} T^{23} + 3153596 p^{9} T^{24} - 253607 p^{10} T^{25} + 34068 p^{11} T^{26} - 2316 p^{12} T^{27} + 269 p^{13} T^{28} - 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + 16 T + 460 T^{2} + 6182 T^{3} + 103105 T^{4} + 1195645 T^{5} + 15130362 T^{6} + 154307107 T^{7} + 1634434125 T^{8} + 14876354705 T^{9} + 137984024424 T^{10} + 1134183253390 T^{11} + 9424749493183 T^{12} + 70531837481534 T^{13} + 531469788689268 T^{14} + 3633615760597546 T^{15} + 531469788689268 p T^{16} + 70531837481534 p^{2} T^{17} + 9424749493183 p^{3} T^{18} + 1134183253390 p^{4} T^{19} + 137984024424 p^{5} T^{20} + 14876354705 p^{6} T^{21} + 1634434125 p^{7} T^{22} + 154307107 p^{8} T^{23} + 15130362 p^{9} T^{24} + 1195645 p^{10} T^{25} + 103105 p^{11} T^{26} + 6182 p^{12} T^{27} + 460 p^{13} T^{28} + 16 p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 - 33 T + 863 T^{2} - 15722 T^{3} + 246347 T^{4} - 3183252 T^{5} + 36912998 T^{6} - 375975233 T^{7} + 3571763601 T^{8} - 31328295324 T^{9} + 268476858640 T^{10} - 2219388789806 T^{11} + 18375669724390 T^{12} - 146996823158855 T^{13} + 1154099940434404 T^{14} - 8551045524129902 T^{15} + 1154099940434404 p T^{16} - 146996823158855 p^{2} T^{17} + 18375669724390 p^{3} T^{18} - 2219388789806 p^{4} T^{19} + 268476858640 p^{5} T^{20} - 31328295324 p^{6} T^{21} + 3571763601 p^{7} T^{22} - 375975233 p^{8} T^{23} + 36912998 p^{9} T^{24} - 3183252 p^{10} T^{25} + 246347 p^{11} T^{26} - 15722 p^{12} T^{27} + 863 p^{13} T^{28} - 33 p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 + 10 T + 438 T^{2} + 3184 T^{3} + 90647 T^{4} + 466961 T^{5} + 11795042 T^{6} + 34131003 T^{7} + 1053566807 T^{8} - 72516125 T^{9} + 66451314548 T^{10} - 316500930128 T^{11} + 3005689076311 T^{12} - 40055060945190 T^{13} + 114526107347490 T^{14} - 2931515365957526 T^{15} + 114526107347490 p T^{16} - 40055060945190 p^{2} T^{17} + 3005689076311 p^{3} T^{18} - 316500930128 p^{4} T^{19} + 66451314548 p^{5} T^{20} - 72516125 p^{6} T^{21} + 1053566807 p^{7} T^{22} + 34131003 p^{8} T^{23} + 11795042 p^{9} T^{24} + 466961 p^{10} T^{25} + 90647 p^{11} T^{26} + 3184 p^{12} T^{27} + 438 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 - T + 523 T^{2} - 129 T^{3} + 136932 T^{4} + 51712 T^{5} + 23976824 T^{6} + 20622714 T^{7} + 3155814965 T^{8} + 3736521011 T^{9} + 331449399927 T^{10} + 447634864256 T^{11} + 28677224204791 T^{12} + 39659467357703 T^{13} + 2076688162703281 T^{14} + 2726716307441572 T^{15} + 2076688162703281 p T^{16} + 39659467357703 p^{2} T^{17} + 28677224204791 p^{3} T^{18} + 447634864256 p^{4} T^{19} + 331449399927 p^{5} T^{20} + 3736521011 p^{6} T^{21} + 3155814965 p^{7} T^{22} + 20622714 p^{8} T^{23} + 23976824 p^{9} T^{24} + 51712 p^{10} T^{25} + 136932 p^{11} T^{26} - 129 p^{12} T^{27} + 523 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 - 20 T + 429 T^{2} - 5822 T^{3} + 73632 T^{4} - 722379 T^{5} + 6792071 T^{6} - 53130812 T^{7} + 435608371 T^{8} - 3414484347 T^{9} + 30999954694 T^{10} - 278288722590 T^{11} + 2671766006547 T^{12} - 23401501064390 T^{13} + 206689542780553 T^{14} - 1689647377929920 T^{15} + 206689542780553 p T^{16} - 23401501064390 p^{2} T^{17} + 2671766006547 p^{3} T^{18} - 278288722590 p^{4} T^{19} + 30999954694 p^{5} T^{20} - 3414484347 p^{6} T^{21} + 435608371 p^{7} T^{22} - 53130812 p^{8} T^{23} + 6792071 p^{9} T^{24} - 722379 p^{10} T^{25} + 73632 p^{11} T^{26} - 5822 p^{12} T^{27} + 429 p^{13} T^{28} - 20 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 - 5 T + 673 T^{2} - 3977 T^{3} + 223612 T^{4} - 1485542 T^{5} + 49080265 T^{6} - 348399794 T^{7} + 8003819994 T^{8} - 57819268023 T^{9} + 1028517214635 T^{10} - 7247014954358 T^{11} + 107290099585961 T^{12} - 713261170497997 T^{13} + 9210290678641452 T^{14} - 56332695950174480 T^{15} + 9210290678641452 p T^{16} - 713261170497997 p^{2} T^{17} + 107290099585961 p^{3} T^{18} - 7247014954358 p^{4} T^{19} + 1028517214635 p^{5} T^{20} - 57819268023 p^{6} T^{21} + 8003819994 p^{7} T^{22} - 348399794 p^{8} T^{23} + 49080265 p^{9} T^{24} - 1485542 p^{10} T^{25} + 223612 p^{11} T^{26} - 3977 p^{12} T^{27} + 673 p^{13} T^{28} - 5 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 3 T + 583 T^{2} + 555 T^{3} + 162403 T^{4} - 132936 T^{5} + 29712937 T^{6} - 66363985 T^{7} + 4114280860 T^{8} - 13241903644 T^{9} + 463723711956 T^{10} - 1765474643243 T^{11} + 44185799832094 T^{12} - 179873270172127 T^{13} + 3652540420576382 T^{14} - 14631397178604766 T^{15} + 3652540420576382 p T^{16} - 179873270172127 p^{2} T^{17} + 44185799832094 p^{3} T^{18} - 1765474643243 p^{4} T^{19} + 463723711956 p^{5} T^{20} - 13241903644 p^{6} T^{21} + 4114280860 p^{7} T^{22} - 66363985 p^{8} T^{23} + 29712937 p^{9} T^{24} - 132936 p^{10} T^{25} + 162403 p^{11} T^{26} + 555 p^{12} T^{27} + 583 p^{13} T^{28} + 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 - 25 T + 867 T^{2} - 15594 T^{3} + 324404 T^{4} - 4707655 T^{5} + 75412400 T^{6} - 941871353 T^{7} + 12777037715 T^{8} - 142497901288 T^{9} + 1711757913945 T^{10} - 17371634352920 T^{11} + 188916919425901 T^{12} - 1760893570473008 T^{13} + 17547537073774455 T^{14} - 150880781779786061 T^{15} + 17547537073774455 p T^{16} - 1760893570473008 p^{2} T^{17} + 188916919425901 p^{3} T^{18} - 17371634352920 p^{4} T^{19} + 1711757913945 p^{5} T^{20} - 142497901288 p^{6} T^{21} + 12777037715 p^{7} T^{22} - 941871353 p^{8} T^{23} + 75412400 p^{9} T^{24} - 4707655 p^{10} T^{25} + 324404 p^{11} T^{26} - 15594 p^{12} T^{27} + 867 p^{13} T^{28} - 25 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 - 18 T + 957 T^{2} - 13278 T^{3} + 406532 T^{4} - 4526403 T^{5} + 104640076 T^{6} - 950016557 T^{7} + 18638437942 T^{8} - 138413158803 T^{9} + 2492847506614 T^{10} - 15191914173358 T^{11} + 268012339072371 T^{12} - 1377333735196800 T^{13} + 24729349092135191 T^{14} - 115390514275133166 T^{15} + 24729349092135191 p T^{16} - 1377333735196800 p^{2} T^{17} + 268012339072371 p^{3} T^{18} - 15191914173358 p^{4} T^{19} + 2492847506614 p^{5} T^{20} - 138413158803 p^{6} T^{21} + 18638437942 p^{7} T^{22} - 950016557 p^{8} T^{23} + 104640076 p^{9} T^{24} - 4526403 p^{10} T^{25} + 406532 p^{11} T^{26} - 13278 p^{12} T^{27} + 957 p^{13} T^{28} - 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 11 T + 645 T^{2} + 8338 T^{3} + 217386 T^{4} + 3018655 T^{5} + 52577251 T^{6} + 707493399 T^{7} + 10081479552 T^{8} + 123380688221 T^{9} + 1563746987133 T^{10} + 17312911230666 T^{11} + 2224710276720 p T^{12} + 2021153054738361 T^{13} + 20843092796609928 T^{14} + 197072121359103194 T^{15} + 20843092796609928 p T^{16} + 2021153054738361 p^{2} T^{17} + 2224710276720 p^{4} T^{18} + 17312911230666 p^{4} T^{19} + 1563746987133 p^{5} T^{20} + 123380688221 p^{6} T^{21} + 10081479552 p^{7} T^{22} + 707493399 p^{8} T^{23} + 52577251 p^{9} T^{24} + 3018655 p^{10} T^{25} + 217386 p^{11} T^{26} + 8338 p^{12} T^{27} + 645 p^{13} T^{28} + 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 - 16 T + 870 T^{2} - 13555 T^{3} + 394150 T^{4} - 5742512 T^{5} + 120559588 T^{6} - 1613534841 T^{7} + 27451908702 T^{8} - 335768070768 T^{9} + 50434733130 p T^{10} - 54643508317377 T^{11} + 702367249354834 T^{12} - 7159961181650448 T^{13} + 82579753815804149 T^{14} - 766573984600929574 T^{15} + 82579753815804149 p T^{16} - 7159961181650448 p^{2} T^{17} + 702367249354834 p^{3} T^{18} - 54643508317377 p^{4} T^{19} + 50434733130 p^{6} T^{20} - 335768070768 p^{6} T^{21} + 27451908702 p^{7} T^{22} - 1613534841 p^{8} T^{23} + 120559588 p^{9} T^{24} - 5742512 p^{10} T^{25} + 394150 p^{11} T^{26} - 13555 p^{12} T^{27} + 870 p^{13} T^{28} - 16 p^{14} T^{29} + p^{15} T^{30} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.87180961663245823767535219679, −1.73079340007142669518234532208, −1.65493564086321759409824520635, −1.62849833859940972570829587107, −1.61208723723443628143101840840, −1.61076436083950247559149220916, −1.57989205777733924488774034894, −1.56901404098607660180193880952, −1.46455624237682239063215166917, −1.32503772785823084353900318680, −1.11903954385246266853464908222, −1.10473739950142081428321693002, −1.09869403552097888024626087514, −0.892726985270966094150608044726, −0.827211650455433768624104700014, −0.790362379969394607797797583562, −0.75223864219943322258050074137, −0.72695519637306173515390691407, −0.68365212194153441821987358470, −0.61908719078495041650121021349, −0.50647880441920067437649780952, −0.50388665197562363794364544963, −0.44737255001197887266123154107, −0.40745626347520622290016569799, −0.15840140354290704407227158277, 0.15840140354290704407227158277, 0.40745626347520622290016569799, 0.44737255001197887266123154107, 0.50388665197562363794364544963, 0.50647880441920067437649780952, 0.61908719078495041650121021349, 0.68365212194153441821987358470, 0.72695519637306173515390691407, 0.75223864219943322258050074137, 0.790362379969394607797797583562, 0.827211650455433768624104700014, 0.892726985270966094150608044726, 1.09869403552097888024626087514, 1.10473739950142081428321693002, 1.11903954385246266853464908222, 1.32503772785823084353900318680, 1.46455624237682239063215166917, 1.56901404098607660180193880952, 1.57989205777733924488774034894, 1.61076436083950247559149220916, 1.61208723723443628143101840840, 1.62849833859940972570829587107, 1.65493564086321759409824520635, 1.73079340007142669518234532208, 1.87180961663245823767535219679
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.