Dirichlet series
| L(s) = 1 | − 3·3-s − 2·5-s − 8·7-s − 6·9-s − 9·11-s − 13-s + 6·15-s − 27·19-s + 24·21-s − 13·23-s − 20·25-s + 25·27-s − 25·31-s + 27·33-s + 16·35-s − 37-s + 3·39-s + 10·41-s − 35·43-s + 12·45-s − 6·47-s + 49-s − 53-s + 18·55-s + 81·57-s − 30·59-s + 3·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s − 3.02·7-s − 2·9-s − 2.71·11-s − 0.277·13-s + 1.54·15-s − 6.19·19-s + 5.23·21-s − 2.71·23-s − 4·25-s + 4.81·27-s − 4.49·31-s + 4.70·33-s + 2.70·35-s − 0.164·37-s + 0.480·39-s + 1.56·41-s − 5.33·43-s + 1.78·45-s − 0.875·47-s + 1/7·49-s − 0.137·53-s + 2.42·55-s + 10.7·57-s − 3.90·59-s + 0.384·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 251^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 251^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{56} \cdot 251^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.21616\times 10^{21}\) |
| Root analytic conductor: | \(5.66285\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 2^{56} \cdot 251^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 251 | \( ( 1 - T )^{14} \) | |
| good | 3 | \( 1 + p T + 5 p T^{2} + 38 T^{3} + 121 T^{4} + 82 p T^{5} + 196 p T^{6} + 943 T^{7} + 568 p T^{8} + 554 p T^{9} + 1559 T^{10} - 3178 T^{11} - 10199 T^{12} - 30997 T^{13} - 51382 T^{14} - 30997 p T^{15} - 10199 p^{2} T^{16} - 3178 p^{3} T^{17} + 1559 p^{4} T^{18} + 554 p^{6} T^{19} + 568 p^{7} T^{20} + 943 p^{7} T^{21} + 196 p^{9} T^{22} + 82 p^{10} T^{23} + 121 p^{10} T^{24} + 38 p^{11} T^{25} + 5 p^{13} T^{26} + p^{14} T^{27} + p^{14} T^{28} \) |
| 5 | \( 1 + 2 T + 24 T^{2} + 59 T^{3} + 64 p T^{4} + 863 T^{5} + 3339 T^{6} + 8624 T^{7} + 28921 T^{8} + 68243 T^{9} + 207217 T^{10} + 458259 T^{11} + 1263673 T^{12} + 2658519 T^{13} + 6736137 T^{14} + 2658519 p T^{15} + 1263673 p^{2} T^{16} + 458259 p^{3} T^{17} + 207217 p^{4} T^{18} + 68243 p^{5} T^{19} + 28921 p^{6} T^{20} + 8624 p^{7} T^{21} + 3339 p^{8} T^{22} + 863 p^{9} T^{23} + 64 p^{11} T^{24} + 59 p^{11} T^{25} + 24 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 7 | \( 1 + 8 T + 9 p T^{2} + 299 T^{3} + 202 p T^{4} + 4980 T^{5} + 18363 T^{6} + 54402 T^{7} + 178145 T^{8} + 485063 T^{9} + 1521945 T^{10} + 4020493 T^{11} + 12333269 T^{12} + 31560379 T^{13} + 91881835 T^{14} + 31560379 p T^{15} + 12333269 p^{2} T^{16} + 4020493 p^{3} T^{17} + 1521945 p^{4} T^{18} + 485063 p^{5} T^{19} + 178145 p^{6} T^{20} + 54402 p^{7} T^{21} + 18363 p^{8} T^{22} + 4980 p^{9} T^{23} + 202 p^{11} T^{24} + 299 p^{11} T^{25} + 9 p^{13} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 + 9 T + 100 T^{2} + 687 T^{3} + 444 p T^{4} + 27546 T^{5} + 155686 T^{6} + 756862 T^{7} + 3649790 T^{8} + 15719595 T^{9} + 66852900 T^{10} + 258836085 T^{11} + 987952117 T^{12} + 3464423852 T^{13} + 11971213668 T^{14} + 3464423852 p T^{15} + 987952117 p^{2} T^{16} + 258836085 p^{3} T^{17} + 66852900 p^{4} T^{18} + 15719595 p^{5} T^{19} + 3649790 p^{6} T^{20} + 756862 p^{7} T^{21} + 155686 p^{8} T^{22} + 27546 p^{9} T^{23} + 444 p^{11} T^{24} + 687 p^{11} T^{25} + 100 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( 1 + T + 77 T^{2} + 118 T^{3} + 3157 T^{4} + 5386 T^{5} + 7130 p T^{6} + 150851 T^{7} + 2123696 T^{8} + 3170308 T^{9} + 39751641 T^{10} + 54234088 T^{11} + 633863833 T^{12} + 790255061 T^{13} + 8807951688 T^{14} + 790255061 p T^{15} + 633863833 p^{2} T^{16} + 54234088 p^{3} T^{17} + 39751641 p^{4} T^{18} + 3170308 p^{5} T^{19} + 2123696 p^{6} T^{20} + 150851 p^{7} T^{21} + 7130 p^{9} T^{22} + 5386 p^{9} T^{23} + 3157 p^{10} T^{24} + 118 p^{11} T^{25} + 77 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 104 T^{2} + 19 T^{3} + 5532 T^{4} + 2047 T^{5} + 205715 T^{6} + 93642 T^{7} + 6080575 T^{8} + 2798407 T^{9} + 150481407 T^{10} + 68169151 T^{11} + 3176734263 T^{12} + 1420430387 T^{13} + 57926949651 T^{14} + 1420430387 p T^{15} + 3176734263 p^{2} T^{16} + 68169151 p^{3} T^{17} + 150481407 p^{4} T^{18} + 2798407 p^{5} T^{19} + 6080575 p^{6} T^{20} + 93642 p^{7} T^{21} + 205715 p^{8} T^{22} + 2047 p^{9} T^{23} + 5532 p^{10} T^{24} + 19 p^{11} T^{25} + 104 p^{12} T^{26} + p^{14} T^{28} \) | |
| 19 | \( 1 + 27 T + 441 T^{2} + 5254 T^{3} + 50682 T^{4} + 21825 p T^{5} + 2979448 T^{6} + 19196965 T^{7} + 113021852 T^{8} + 616541144 T^{9} + 3160784719 T^{10} + 15400093521 T^{11} + 72100535921 T^{12} + 326801388214 T^{13} + 1442586078560 T^{14} + 326801388214 p T^{15} + 72100535921 p^{2} T^{16} + 15400093521 p^{3} T^{17} + 3160784719 p^{4} T^{18} + 616541144 p^{5} T^{19} + 113021852 p^{6} T^{20} + 19196965 p^{7} T^{21} + 2979448 p^{8} T^{22} + 21825 p^{10} T^{23} + 50682 p^{10} T^{24} + 5254 p^{11} T^{25} + 441 p^{12} T^{26} + 27 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 + 13 T + 190 T^{2} + 66 p T^{3} + 13387 T^{4} + 77351 T^{5} + 513478 T^{6} + 2186676 T^{7} + 11981481 T^{8} + 31447409 T^{9} + 146114163 T^{10} - 206892093 T^{11} - 842453049 T^{12} - 23183395277 T^{13} - 2884026841 p T^{14} - 23183395277 p T^{15} - 842453049 p^{2} T^{16} - 206892093 p^{3} T^{17} + 146114163 p^{4} T^{18} + 31447409 p^{5} T^{19} + 11981481 p^{6} T^{20} + 2186676 p^{7} T^{21} + 513478 p^{8} T^{22} + 77351 p^{9} T^{23} + 13387 p^{10} T^{24} + 66 p^{12} T^{25} + 190 p^{12} T^{26} + 13 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 + 157 T^{2} + 192 T^{3} + 13404 T^{4} + 26074 T^{5} + 840016 T^{6} + 1985546 T^{7} + 1454718 p T^{8} + 108389824 T^{9} + 1784428139 T^{10} + 4581335888 T^{11} + 64702247509 T^{12} + 158910539972 T^{13} + 2018754156720 T^{14} + 158910539972 p T^{15} + 64702247509 p^{2} T^{16} + 4581335888 p^{3} T^{17} + 1784428139 p^{4} T^{18} + 108389824 p^{5} T^{19} + 1454718 p^{7} T^{20} + 1985546 p^{7} T^{21} + 840016 p^{8} T^{22} + 26074 p^{9} T^{23} + 13404 p^{10} T^{24} + 192 p^{11} T^{25} + 157 p^{12} T^{26} + p^{14} T^{28} \) | |
| 31 | \( 1 + 25 T + 503 T^{2} + 6946 T^{3} + 83553 T^{4} + 823630 T^{5} + 7364616 T^{6} + 57517098 T^{7} + 419903137 T^{8} + 2780464125 T^{9} + 17714198221 T^{10} + 105465364251 T^{11} + 621052204731 T^{12} + 3495835300725 T^{13} + 19820760784583 T^{14} + 3495835300725 p T^{15} + 621052204731 p^{2} T^{16} + 105465364251 p^{3} T^{17} + 17714198221 p^{4} T^{18} + 2780464125 p^{5} T^{19} + 419903137 p^{6} T^{20} + 57517098 p^{7} T^{21} + 7364616 p^{8} T^{22} + 823630 p^{9} T^{23} + 83553 p^{10} T^{24} + 6946 p^{11} T^{25} + 503 p^{12} T^{26} + 25 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 + T + 206 T^{2} + 417 T^{3} + 22954 T^{4} + 67484 T^{5} + 1814134 T^{6} + 6641756 T^{7} + 115182508 T^{8} + 460182755 T^{9} + 6224084362 T^{10} + 24574724091 T^{11} + 290341422433 T^{12} + 1073562144200 T^{13} + 11624734096484 T^{14} + 1073562144200 p T^{15} + 290341422433 p^{2} T^{16} + 24574724091 p^{3} T^{17} + 6224084362 p^{4} T^{18} + 460182755 p^{5} T^{19} + 115182508 p^{6} T^{20} + 6641756 p^{7} T^{21} + 1814134 p^{8} T^{22} + 67484 p^{9} T^{23} + 22954 p^{10} T^{24} + 417 p^{11} T^{25} + 206 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 - 10 T + 271 T^{2} - 2287 T^{3} + 35114 T^{4} - 272478 T^{5} + 3121403 T^{6} - 22996332 T^{7} + 218558009 T^{8} - 1529053321 T^{9} + 12761407637 T^{10} - 84350370261 T^{11} + 639294414953 T^{12} - 3977962213993 T^{13} + 27952401522009 T^{14} - 3977962213993 p T^{15} + 639294414953 p^{2} T^{16} - 84350370261 p^{3} T^{17} + 12761407637 p^{4} T^{18} - 1529053321 p^{5} T^{19} + 218558009 p^{6} T^{20} - 22996332 p^{7} T^{21} + 3121403 p^{8} T^{22} - 272478 p^{9} T^{23} + 35114 p^{10} T^{24} - 2287 p^{11} T^{25} + 271 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 + 35 T + 948 T^{2} + 18609 T^{3} + 311924 T^{4} + 4456696 T^{5} + 56842554 T^{6} + 650599032 T^{7} + 6808036286 T^{8} + 65426928661 T^{9} + 582811610820 T^{10} + 4824699695991 T^{11} + 37308717328405 T^{12} + 269691818544208 T^{13} + 1827487675896028 T^{14} + 269691818544208 p T^{15} + 37308717328405 p^{2} T^{16} + 4824699695991 p^{3} T^{17} + 582811610820 p^{4} T^{18} + 65426928661 p^{5} T^{19} + 6808036286 p^{6} T^{20} + 650599032 p^{7} T^{21} + 56842554 p^{8} T^{22} + 4456696 p^{9} T^{23} + 311924 p^{10} T^{24} + 18609 p^{11} T^{25} + 948 p^{12} T^{26} + 35 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 + 6 T + 382 T^{2} + 1059 T^{3} + 61842 T^{4} - 36571 T^{5} + 5929810 T^{6} - 26059681 T^{7} + 432153524 T^{8} - 3250819243 T^{9} + 30303276714 T^{10} - 230841065832 T^{11} + 2036943202521 T^{12} - 12145587908550 T^{13} + 110601003501836 T^{14} - 12145587908550 p T^{15} + 2036943202521 p^{2} T^{16} - 230841065832 p^{3} T^{17} + 30303276714 p^{4} T^{18} - 3250819243 p^{5} T^{19} + 432153524 p^{6} T^{20} - 26059681 p^{7} T^{21} + 5929810 p^{8} T^{22} - 36571 p^{9} T^{23} + 61842 p^{10} T^{24} + 1059 p^{11} T^{25} + 382 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 + T + 336 T^{2} + 657 T^{3} + 55960 T^{4} + 176338 T^{5} + 6430150 T^{6} + 27382410 T^{7} + 584794578 T^{8} + 2927367915 T^{9} + 44477723032 T^{10} + 239086649523 T^{11} + 2895999634685 T^{12} + 15648142913420 T^{13} + 164010413112036 T^{14} + 15648142913420 p T^{15} + 2895999634685 p^{2} T^{16} + 239086649523 p^{3} T^{17} + 44477723032 p^{4} T^{18} + 2927367915 p^{5} T^{19} + 584794578 p^{6} T^{20} + 27382410 p^{7} T^{21} + 6430150 p^{8} T^{22} + 176338 p^{9} T^{23} + 55960 p^{10} T^{24} + 657 p^{11} T^{25} + 336 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 30 T + 779 T^{2} + 13424 T^{3} + 209000 T^{4} + 2609668 T^{5} + 30561096 T^{6} + 309458644 T^{7} + 3067074882 T^{8} + 27703864660 T^{9} + 254359244333 T^{10} + 2182190578586 T^{11} + 19036964650525 T^{12} + 153468834699824 T^{13} + 1232796925002464 T^{14} + 153468834699824 p T^{15} + 19036964650525 p^{2} T^{16} + 2182190578586 p^{3} T^{17} + 254359244333 p^{4} T^{18} + 27703864660 p^{5} T^{19} + 3067074882 p^{6} T^{20} + 309458644 p^{7} T^{21} + 30561096 p^{8} T^{22} + 2609668 p^{9} T^{23} + 209000 p^{10} T^{24} + 13424 p^{11} T^{25} + 779 p^{12} T^{26} + 30 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 3 T + 403 T^{2} - 1236 T^{3} + 82870 T^{4} - 256913 T^{5} + 11777036 T^{6} - 35236793 T^{7} + 1301617432 T^{8} - 3621165890 T^{9} + 118008463253 T^{10} - 302860230171 T^{11} + 9025363533113 T^{12} - 21515936469586 T^{13} + 592385566983864 T^{14} - 21515936469586 p T^{15} + 9025363533113 p^{2} T^{16} - 302860230171 p^{3} T^{17} + 118008463253 p^{4} T^{18} - 3621165890 p^{5} T^{19} + 1301617432 p^{6} T^{20} - 35236793 p^{7} T^{21} + 11777036 p^{8} T^{22} - 256913 p^{9} T^{23} + 82870 p^{10} T^{24} - 1236 p^{11} T^{25} + 403 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 + 22 T + 694 T^{2} + 10439 T^{3} + 193492 T^{4} + 34113 p T^{5} + 32492463 T^{6} + 328286398 T^{7} + 3998651363 T^{8} + 36396393149 T^{9} + 399795726501 T^{10} + 3357364823787 T^{11} + 33894768425127 T^{12} + 263865853266971 T^{13} + 2457414462651723 T^{14} + 263865853266971 p T^{15} + 33894768425127 p^{2} T^{16} + 3357364823787 p^{3} T^{17} + 399795726501 p^{4} T^{18} + 36396393149 p^{5} T^{19} + 3998651363 p^{6} T^{20} + 328286398 p^{7} T^{21} + 32492463 p^{8} T^{22} + 34113 p^{10} T^{23} + 193492 p^{10} T^{24} + 10439 p^{11} T^{25} + 694 p^{12} T^{26} + 22 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 6 T + 440 T^{2} + 951 T^{3} + 96092 T^{4} - 58657 T^{5} + 14918598 T^{6} - 40123203 T^{7} + 1874503078 T^{8} - 7242827855 T^{9} + 200997151216 T^{10} - 842990815480 T^{11} + 18428438400077 T^{12} - 74289209148314 T^{13} + 1428760173206436 T^{14} - 74289209148314 p T^{15} + 18428438400077 p^{2} T^{16} - 842990815480 p^{3} T^{17} + 200997151216 p^{4} T^{18} - 7242827855 p^{5} T^{19} + 1874503078 p^{6} T^{20} - 40123203 p^{7} T^{21} + 14918598 p^{8} T^{22} - 58657 p^{9} T^{23} + 96092 p^{10} T^{24} + 951 p^{11} T^{25} + 440 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( 1 - 5 T + 509 T^{2} - 2246 T^{3} + 131047 T^{4} - 543096 T^{5} + 22785570 T^{6} - 92847232 T^{7} + 3016491029 T^{8} - 12396170995 T^{9} + 324499520677 T^{10} - 1340853514487 T^{11} + 29471947693033 T^{12} - 118946765135011 T^{13} + 2308572431873909 T^{14} - 118946765135011 p T^{15} + 29471947693033 p^{2} T^{16} - 1340853514487 p^{3} T^{17} + 324499520677 p^{4} T^{18} - 12396170995 p^{5} T^{19} + 3016491029 p^{6} T^{20} - 92847232 p^{7} T^{21} + 22785570 p^{8} T^{22} - 543096 p^{9} T^{23} + 131047 p^{10} T^{24} - 2246 p^{11} T^{25} + 509 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 + 56 T + 2178 T^{2} + 61573 T^{3} + 1444638 T^{4} + 28678155 T^{5} + 502092413 T^{6} + 7837978292 T^{7} + 111253380003 T^{8} + 1445274307857 T^{9} + 17375055426959 T^{10} + 193997565121459 T^{11} + 2024946369538101 T^{12} + 19782596308792413 T^{13} + 181527502415863181 T^{14} + 19782596308792413 p T^{15} + 2024946369538101 p^{2} T^{16} + 193997565121459 p^{3} T^{17} + 17375055426959 p^{4} T^{18} + 1445274307857 p^{5} T^{19} + 111253380003 p^{6} T^{20} + 7837978292 p^{7} T^{21} + 502092413 p^{8} T^{22} + 28678155 p^{9} T^{23} + 1444638 p^{10} T^{24} + 61573 p^{11} T^{25} + 2178 p^{12} T^{26} + 56 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 28 T + 1194 T^{2} - 24258 T^{3} + 597660 T^{4} - 9577457 T^{5} + 174046785 T^{6} - 2306100582 T^{7} + 33960396380 T^{8} - 384505903734 T^{9} + 4841020526267 T^{10} - 48162250372593 T^{11} + 538120608580119 T^{12} - 4819591768627718 T^{13} + 48926682521657972 T^{14} - 4819591768627718 p T^{15} + 538120608580119 p^{2} T^{16} - 48162250372593 p^{3} T^{17} + 4841020526267 p^{4} T^{18} - 384505903734 p^{5} T^{19} + 33960396380 p^{6} T^{20} - 2306100582 p^{7} T^{21} + 174046785 p^{8} T^{22} - 9577457 p^{9} T^{23} + 597660 p^{10} T^{24} - 24258 p^{11} T^{25} + 1194 p^{12} T^{26} - 28 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 + 24 T + 669 T^{2} + 10517 T^{3} + 185238 T^{4} + 2263722 T^{5} + 31585183 T^{6} + 325130743 T^{7} + 4027076643 T^{8} + 37730303200 T^{9} + 449488705240 T^{10} + 4053222457673 T^{11} + 47081758898733 T^{12} + 4583137834810 p T^{13} + 4474499931731100 T^{14} + 4583137834810 p^{2} T^{15} + 47081758898733 p^{2} T^{16} + 4053222457673 p^{3} T^{17} + 449488705240 p^{4} T^{18} + 37730303200 p^{5} T^{19} + 4027076643 p^{6} T^{20} + 325130743 p^{7} T^{21} + 31585183 p^{8} T^{22} + 2263722 p^{9} T^{23} + 185238 p^{10} T^{24} + 10517 p^{11} T^{25} + 669 p^{12} T^{26} + 24 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 - 6 T + 832 T^{2} - 5918 T^{3} + 341876 T^{4} - 2755486 T^{5} + 92593566 T^{6} - 810972134 T^{7} + 18572514318 T^{8} - 169569347650 T^{9} + 2931193224296 T^{10} - 26741162517482 T^{11} + 376439434844093 T^{12} - 3283902645684332 T^{13} + 40023991828359028 T^{14} - 3283902645684332 p T^{15} + 376439434844093 p^{2} T^{16} - 26741162517482 p^{3} T^{17} + 2931193224296 p^{4} T^{18} - 169569347650 p^{5} T^{19} + 18572514318 p^{6} T^{20} - 810972134 p^{7} T^{21} + 92593566 p^{8} T^{22} - 2755486 p^{9} T^{23} + 341876 p^{10} T^{24} - 5918 p^{11} T^{25} + 832 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.67421531497947311277215939792, −2.58489305920504541511193965858, −2.52758030446072320541524843260, −2.46268680656422246334899328340, −2.46187469539852635216352689161, −2.45482680997341606338609527631, −2.45342970409765551670373553687, −2.41946668116275427057694255022, −2.19993376170803161592352059055, −2.18222004965771762916260299711, −2.03735645034927012588691436541, −1.97977689628397592383325537243, −1.90335421010329602921815275212, −1.80714027234028779196041121164, −1.61393202819819530068506581777, −1.60077071308458554953123399649, −1.59973085444631968459304310728, −1.58165033695784256171574628398, −1.33036477343944446730120631143, −1.29031661089145486074548726307, −1.19847794935425772472988965078, −1.18885970305217822492627354000, −1.13785571757702803400160570813, −1.13514857787726696152637456756, −0.74149099008737844816316571595, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.74149099008737844816316571595, 1.13514857787726696152637456756, 1.13785571757702803400160570813, 1.18885970305217822492627354000, 1.19847794935425772472988965078, 1.29031661089145486074548726307, 1.33036477343944446730120631143, 1.58165033695784256171574628398, 1.59973085444631968459304310728, 1.60077071308458554953123399649, 1.61393202819819530068506581777, 1.80714027234028779196041121164, 1.90335421010329602921815275212, 1.97977689628397592383325537243, 2.03735645034927012588691436541, 2.18222004965771762916260299711, 2.19993376170803161592352059055, 2.41946668116275427057694255022, 2.45342970409765551670373553687, 2.45482680997341606338609527631, 2.46187469539852635216352689161, 2.46268680656422246334899328340, 2.52758030446072320541524843260, 2.58489305920504541511193965858, 2.67421531497947311277215939792
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.