Dirichlet series
| L(s) = 1 | − 10·4-s + 7-s − 4·8-s + 2·11-s + 5·13-s + 47·16-s − 4·17-s − 11·19-s + 8·23-s + 21·25-s − 10·28-s − 15·29-s + 3·31-s + 46·32-s − 8·37-s − 19·41-s + 11·43-s − 20·44-s − 5·47-s + 4·49-s − 50·52-s − 36·53-s − 4·56-s − 34·59-s − 22·61-s − 136·64-s + 26·67-s + ⋯ |
| L(s) = 1 | − 5·4-s + 0.377·7-s − 1.41·8-s + 0.603·11-s + 1.38·13-s + 47/4·16-s − 0.970·17-s − 2.52·19-s + 1.66·23-s + 21/5·25-s − 1.88·28-s − 2.78·29-s + 0.538·31-s + 8.13·32-s − 1.31·37-s − 2.96·41-s + 1.67·43-s − 3.01·44-s − 0.729·47-s + 4/7·49-s − 6.93·52-s − 4.94·53-s − 0.534·56-s − 4.42·59-s − 2.81·61-s − 17·64-s + 3.17·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11937\times 10^{13}\) |
| Root analytic conductor: | \(2.55729\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.009685207624\) |
| \(L(\frac12)\) | \(\approx\) | \(0.009685207624\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T - 3 T^{2} + 2 T^{3} + 34 T^{4} - 122 T^{6} - 565 T^{7} + 3111 T^{8} - 565 p T^{9} - 122 p^{2} T^{10} + 34 p^{4} T^{12} + 2 p^{5} T^{13} - 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 6 T^{2} + 10 T^{3} + 19 p T^{4} - 720 T^{5} + 145 T^{6} - 4565 T^{7} + 64050 T^{8} - 4565 p T^{9} + 145 p^{2} T^{10} - 720 p^{3} T^{11} + 19 p^{5} T^{12} + 10 p^{5} T^{13} + 6 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( ( 1 + 5 T^{2} + p T^{3} + 7 p T^{4} + 7 T^{5} + 17 p T^{6} + 15 T^{7} + 73 T^{8} + 15 p T^{9} + 17 p^{3} T^{10} + 7 p^{3} T^{11} + 7 p^{5} T^{12} + p^{6} T^{13} + 5 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( 1 - 21 T^{2} - 2 p T^{3} + 197 T^{4} + 163 T^{5} - 239 p T^{6} - 743 T^{7} + 6997 T^{8} - 33 p T^{9} - 10259 p T^{10} - 4408 T^{11} + 340012 T^{12} + 149669 T^{13} - 327916 p T^{14} - 505874 T^{15} + 7287304 T^{16} - 505874 p T^{17} - 327916 p^{3} T^{18} + 149669 p^{3} T^{19} + 340012 p^{4} T^{20} - 4408 p^{5} T^{21} - 10259 p^{7} T^{22} - 33 p^{8} T^{23} + 6997 p^{8} T^{24} - 743 p^{9} T^{25} - 239 p^{11} T^{26} + 163 p^{11} T^{27} + 197 p^{12} T^{28} - 2 p^{14} T^{29} - 21 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 2 T - 43 T^{2} + 128 T^{3} + 751 T^{4} - 2911 T^{5} - 7079 T^{6} + 25011 T^{7} + 89586 T^{8} + 16491 T^{9} - 2184437 T^{10} - 146331 T^{11} + 34120480 T^{12} - 33194813 T^{13} - 280856670 T^{14} + 275502867 T^{15} + 1982899103 T^{16} + 275502867 p T^{17} - 280856670 p^{2} T^{18} - 33194813 p^{3} T^{19} + 34120480 p^{4} T^{20} - 146331 p^{5} T^{21} - 2184437 p^{6} T^{22} + 16491 p^{7} T^{23} + 89586 p^{8} T^{24} + 25011 p^{9} T^{25} - 7079 p^{10} T^{26} - 2911 p^{11} T^{27} + 751 p^{12} T^{28} + 128 p^{13} T^{29} - 43 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 2 T + 4 p T^{2} + 146 T^{3} + 147 p T^{4} + 5560 T^{5} + 63345 T^{6} + 135302 T^{7} + 1216691 T^{8} + 135302 p T^{9} + 63345 p^{2} T^{10} + 5560 p^{3} T^{11} + 147 p^{5} T^{12} + 146 p^{5} T^{13} + 4 p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 11 T - p T^{2} - 422 T^{3} + 1327 T^{4} + 13231 T^{5} - 52770 T^{6} - 286830 T^{7} + 1316435 T^{8} + 3604514 T^{9} - 28081056 T^{10} - 2271598 p T^{11} + 436051189 T^{12} + 420246053 T^{13} - 6706056535 T^{14} + 224221973 T^{15} + 142211799535 T^{16} + 224221973 p T^{17} - 6706056535 p^{2} T^{18} + 420246053 p^{3} T^{19} + 436051189 p^{4} T^{20} - 2271598 p^{6} T^{21} - 28081056 p^{6} T^{22} + 3604514 p^{7} T^{23} + 1316435 p^{8} T^{24} - 286830 p^{9} T^{25} - 52770 p^{10} T^{26} + 13231 p^{11} T^{27} + 1327 p^{12} T^{28} - 422 p^{13} T^{29} - p^{15} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 4 T + 134 T^{2} - 486 T^{3} + 8278 T^{4} - 27515 T^{5} + 13874 p T^{6} - 953642 T^{7} + 8621489 T^{8} - 953642 p T^{9} + 13874 p^{3} T^{10} - 27515 p^{3} T^{11} + 8278 p^{4} T^{12} - 486 p^{5} T^{13} + 134 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 15 T - 6 T^{2} - 815 T^{3} + 2345 T^{4} + 56626 T^{5} - 990 p T^{6} - 61073 p T^{7} + 2048957 T^{8} + 60143926 T^{9} + 115849537 T^{10} - 850143313 T^{11} - 6898778608 T^{12} + 21046980820 T^{13} + 452641755556 T^{14} + 138544395442 T^{15} - 12857101724960 T^{16} + 138544395442 p T^{17} + 452641755556 p^{2} T^{18} + 21046980820 p^{3} T^{19} - 6898778608 p^{4} T^{20} - 850143313 p^{5} T^{21} + 115849537 p^{6} T^{22} + 60143926 p^{7} T^{23} + 2048957 p^{8} T^{24} - 61073 p^{10} T^{25} - 990 p^{11} T^{26} + 56626 p^{11} T^{27} + 2345 p^{12} T^{28} - 815 p^{13} T^{29} - 6 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 3 T - 80 T^{2} + 753 T^{3} + 1092 T^{4} - 46457 T^{5} + 220332 T^{6} + 1046342 T^{7} - 14211319 T^{8} + 31888447 T^{9} + 369460130 T^{10} - 2912434243 T^{11} + 1186850305 T^{12} + 95815568717 T^{13} - 439310068799 T^{14} - 1217655157396 T^{15} + 18735868674340 T^{16} - 1217655157396 p T^{17} - 439310068799 p^{2} T^{18} + 95815568717 p^{3} T^{19} + 1186850305 p^{4} T^{20} - 2912434243 p^{5} T^{21} + 369460130 p^{6} T^{22} + 31888447 p^{7} T^{23} - 14211319 p^{8} T^{24} + 1046342 p^{9} T^{25} + 220332 p^{10} T^{26} - 46457 p^{11} T^{27} + 1092 p^{12} T^{28} + 753 p^{13} T^{29} - 80 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 4 T + 177 T^{2} + 19 p T^{3} + 15864 T^{4} + 58980 T^{5} + 951780 T^{6} + 3131831 T^{7} + 41176929 T^{8} + 3131831 p T^{9} + 951780 p^{2} T^{10} + 58980 p^{3} T^{11} + 15864 p^{4} T^{12} + 19 p^{6} T^{13} + 177 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 + 19 T + 55 T^{2} - 898 T^{3} - 5698 T^{4} - 16130 T^{5} - 337390 T^{6} - 2358667 T^{7} + 9439631 T^{8} + 161858061 T^{9} + 581239977 T^{10} + 3187530408 T^{11} + 37046289522 T^{12} + 51167015014 T^{13} - 1547466225765 T^{14} - 9408233383915 T^{15} - 34873463295586 T^{16} - 9408233383915 p T^{17} - 1547466225765 p^{2} T^{18} + 51167015014 p^{3} T^{19} + 37046289522 p^{4} T^{20} + 3187530408 p^{5} T^{21} + 581239977 p^{6} T^{22} + 161858061 p^{7} T^{23} + 9439631 p^{8} T^{24} - 2358667 p^{9} T^{25} - 337390 p^{10} T^{26} - 16130 p^{11} T^{27} - 5698 p^{12} T^{28} - 898 p^{13} T^{29} + 55 p^{14} T^{30} + 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 11 T - 135 T^{2} + 2100 T^{3} + 6137 T^{4} - 176179 T^{5} - 40953 T^{6} + 8402878 T^{7} + 3448032 T^{8} - 271175414 T^{9} - 1428475026 T^{10} + 9515158664 T^{11} + 124800942246 T^{12} - 411307083359 T^{13} - 6208904645339 T^{14} + 198702541525 p T^{15} + 260553692815250 T^{16} + 198702541525 p^{2} T^{17} - 6208904645339 p^{2} T^{18} - 411307083359 p^{3} T^{19} + 124800942246 p^{4} T^{20} + 9515158664 p^{5} T^{21} - 1428475026 p^{6} T^{22} - 271175414 p^{7} T^{23} + 3448032 p^{8} T^{24} + 8402878 p^{9} T^{25} - 40953 p^{10} T^{26} - 176179 p^{11} T^{27} + 6137 p^{12} T^{28} + 2100 p^{13} T^{29} - 135 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 5 T - 168 T^{2} - 161 T^{3} + 16599 T^{4} - 36967 T^{5} - 885940 T^{6} + 5198908 T^{7} + 21412749 T^{8} - 271776345 T^{9} + 346611318 T^{10} + 4913373621 T^{11} - 20631932251 T^{12} + 180472147091 T^{13} - 1523045035175 T^{14} - 7154378798768 T^{15} + 147263530000230 T^{16} - 7154378798768 p T^{17} - 1523045035175 p^{2} T^{18} + 180472147091 p^{3} T^{19} - 20631932251 p^{4} T^{20} + 4913373621 p^{5} T^{21} + 346611318 p^{6} T^{22} - 271776345 p^{7} T^{23} + 21412749 p^{8} T^{24} + 5198908 p^{9} T^{25} - 885940 p^{10} T^{26} - 36967 p^{11} T^{27} + 16599 p^{12} T^{28} - 161 p^{13} T^{29} - 168 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 36 T + 508 T^{2} + 3932 T^{3} + 29444 T^{4} + 280817 T^{5} + 1836996 T^{6} + 8495258 T^{7} + 63588909 T^{8} + 353688724 T^{9} + 1151048267 T^{10} + 22337541825 T^{11} + 242039231570 T^{12} + 1659435705048 T^{13} + 17509974066650 T^{14} + 155428900722104 T^{15} + 1072759047046030 T^{16} + 155428900722104 p T^{17} + 17509974066650 p^{2} T^{18} + 1659435705048 p^{3} T^{19} + 242039231570 p^{4} T^{20} + 22337541825 p^{5} T^{21} + 1151048267 p^{6} T^{22} + 353688724 p^{7} T^{23} + 63588909 p^{8} T^{24} + 8495258 p^{9} T^{25} + 1836996 p^{10} T^{26} + 280817 p^{11} T^{27} + 29444 p^{12} T^{28} + 3932 p^{13} T^{29} + 508 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 17 T + 406 T^{2} + 4290 T^{3} + 60383 T^{4} + 466365 T^{5} + 5192871 T^{6} + 33146017 T^{7} + 335346919 T^{8} + 33146017 p T^{9} + 5192871 p^{2} T^{10} + 466365 p^{3} T^{11} + 60383 p^{4} T^{12} + 4290 p^{5} T^{13} + 406 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 + 22 T - 95 T^{2} - 3580 T^{3} + 31457 T^{4} + 560563 T^{5} - 4904245 T^{6} - 53746097 T^{7} + 634614646 T^{8} + 3868867413 T^{9} - 63304590509 T^{10} - 211847608419 T^{11} + 5064616862260 T^{12} + 7933710068297 T^{13} - 355599356859504 T^{14} - 168467878784829 T^{15} + 22415396390386199 T^{16} - 168467878784829 p T^{17} - 355599356859504 p^{2} T^{18} + 7933710068297 p^{3} T^{19} + 5064616862260 p^{4} T^{20} - 211847608419 p^{5} T^{21} - 63304590509 p^{6} T^{22} + 3868867413 p^{7} T^{23} + 634614646 p^{8} T^{24} - 53746097 p^{9} T^{25} - 4904245 p^{10} T^{26} + 560563 p^{11} T^{27} + 31457 p^{12} T^{28} - 3580 p^{13} T^{29} - 95 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 26 T + 167 T^{2} + 144 T^{3} + 8449 T^{4} - 117319 T^{5} - 389361 T^{6} - 56323 p T^{7} + 147119946 T^{8} - 250628453 T^{9} + 228999159 T^{10} - 66354454115 T^{11} - 131549954032 T^{12} + 4100116909047 T^{13} + 36001458072338 T^{14} - 214087313935437 T^{15} - 1701481366216205 T^{16} - 214087313935437 p T^{17} + 36001458072338 p^{2} T^{18} + 4100116909047 p^{3} T^{19} - 131549954032 p^{4} T^{20} - 66354454115 p^{5} T^{21} + 228999159 p^{6} T^{22} - 250628453 p^{7} T^{23} + 147119946 p^{8} T^{24} - 56323 p^{10} T^{25} - 389361 p^{10} T^{26} - 117319 p^{11} T^{27} + 8449 p^{12} T^{28} + 144 p^{13} T^{29} + 167 p^{14} T^{30} - 26 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 9 T - 386 T^{2} - 3589 T^{3} + 81235 T^{4} + 746618 T^{5} - 12406552 T^{6} - 106145741 T^{7} + 1537530601 T^{8} + 11338510824 T^{9} - 161456593527 T^{10} - 918377967383 T^{11} + 14822377342250 T^{12} + 52736724869388 T^{13} - 1218705028862170 T^{14} - 1440399689396096 T^{15} + 90831121962753112 T^{16} - 1440399689396096 p T^{17} - 1218705028862170 p^{2} T^{18} + 52736724869388 p^{3} T^{19} + 14822377342250 p^{4} T^{20} - 918377967383 p^{5} T^{21} - 161456593527 p^{6} T^{22} + 11338510824 p^{7} T^{23} + 1537530601 p^{8} T^{24} - 106145741 p^{9} T^{25} - 12406552 p^{10} T^{26} + 746618 p^{11} T^{27} + 81235 p^{12} T^{28} - 3589 p^{13} T^{29} - 386 p^{14} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 6 T - 165 T^{2} + 1806 T^{3} + 26272 T^{4} - 359064 T^{5} + 1422661 T^{6} + 45649262 T^{7} - 424402259 T^{8} + 411144950 T^{9} + 51687211764 T^{10} - 358428082760 T^{11} - 635781335617 T^{12} + 45102010283078 T^{13} - 217262895796719 T^{14} - 1212829928109410 T^{15} + 31435042908856916 T^{16} - 1212829928109410 p T^{17} - 217262895796719 p^{2} T^{18} + 45102010283078 p^{3} T^{19} - 635781335617 p^{4} T^{20} - 358428082760 p^{5} T^{21} + 51687211764 p^{6} T^{22} + 411144950 p^{7} T^{23} - 424402259 p^{8} T^{24} + 45649262 p^{9} T^{25} + 1422661 p^{10} T^{26} - 359064 p^{11} T^{27} + 26272 p^{12} T^{28} + 1806 p^{13} T^{29} - 165 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 16 T - 231 T^{2} + 4320 T^{3} + 34115 T^{4} - 635696 T^{5} - 4129200 T^{6} + 64433872 T^{7} + 434574435 T^{8} - 5166633824 T^{9} - 32859288299 T^{10} + 304450300496 T^{11} + 1757236296274 T^{12} - 12357156907472 T^{13} - 63637790409375 T^{14} + 230917397502624 T^{15} + 3042650590829296 T^{16} + 230917397502624 p T^{17} - 63637790409375 p^{2} T^{18} - 12357156907472 p^{3} T^{19} + 1757236296274 p^{4} T^{20} + 304450300496 p^{5} T^{21} - 32859288299 p^{6} T^{22} - 5166633824 p^{7} T^{23} + 434574435 p^{8} T^{24} + 64433872 p^{9} T^{25} - 4129200 p^{10} T^{26} - 635696 p^{11} T^{27} + 34115 p^{12} T^{28} + 4320 p^{13} T^{29} - 231 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 18 T + 468 T^{2} + 5152 T^{3} + 82570 T^{4} + 680729 T^{5} + 9262051 T^{6} + 65493403 T^{7} + 834546596 T^{8} + 65493403 p T^{9} + 9262051 p^{2} T^{10} + 680729 p^{3} T^{11} + 82570 p^{4} T^{12} + 5152 p^{5} T^{13} + 468 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 20 T + 630 T^{2} - 9755 T^{3} + 178627 T^{4} - 2212339 T^{5} + 29910803 T^{6} - 302486077 T^{7} + 36604201 p T^{8} - 302486077 p T^{9} + 29910803 p^{2} T^{10} - 2212339 p^{3} T^{11} + 178627 p^{4} T^{12} - 9755 p^{5} T^{13} + 630 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 7 T - 310 T^{2} + 2485 T^{3} + 42400 T^{4} - 379400 T^{5} - 3621348 T^{6} + 31983102 T^{7} + 207352950 T^{8} - 1459259835 T^{9} + 6146393957 T^{10} - 52607230120 T^{11} - 3573659402933 T^{12} + 20216002617763 T^{13} + 477664863434550 T^{14} - 1221840745845096 T^{15} - 46591681297798206 T^{16} - 1221840745845096 p T^{17} + 477664863434550 p^{2} T^{18} + 20216002617763 p^{3} T^{19} - 3573659402933 p^{4} T^{20} - 52607230120 p^{5} T^{21} + 6146393957 p^{6} T^{22} - 1459259835 p^{7} T^{23} + 207352950 p^{8} T^{24} + 31983102 p^{9} T^{25} - 3621348 p^{10} T^{26} - 379400 p^{11} T^{27} + 42400 p^{12} T^{28} + 2485 p^{13} T^{29} - 310 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.80937062956901233652855834075, −2.74412336188989428172654756858, −2.61230316777650773774839654536, −2.50131607667951562138488919988, −2.34955615195671736135004871519, −2.25037471218461630076244780905, −2.22672624874443699124917649464, −1.96028355474326836800778077426, −1.90527659181437268732762395719, −1.85750899770044225538748438301, −1.78034405490905666421876820743, −1.77755593261791955218931379751, −1.77425460781925995927203942956, −1.46196051387890084140358705553, −1.35406371550538683406558965348, −1.34337261139963004001069335050, −1.21774800481920246014352171731, −1.13514198303750923600903580682, −0.932859411035934194328855690711, −0.72485473202671129580102995660, −0.69387563448315584187069773466, −0.68631474401703797683411106656, −0.19129626980742535521250403823, −0.10539871373488336726684088014, −0.087260281665605644328103614436, 0.087260281665605644328103614436, 0.10539871373488336726684088014, 0.19129626980742535521250403823, 0.68631474401703797683411106656, 0.69387563448315584187069773466, 0.72485473202671129580102995660, 0.932859411035934194328855690711, 1.13514198303750923600903580682, 1.21774800481920246014352171731, 1.34337261139963004001069335050, 1.35406371550538683406558965348, 1.46196051387890084140358705553, 1.77425460781925995927203942956, 1.77755593261791955218931379751, 1.78034405490905666421876820743, 1.85750899770044225538748438301, 1.90527659181437268732762395719, 1.96028355474326836800778077426, 2.22672624874443699124917649464, 2.25037471218461630076244780905, 2.34955615195671736135004871519, 2.50131607667951562138488919988, 2.61230316777650773774839654536, 2.74412336188989428172654756858, 2.80937062956901233652855834075
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.