| L(s) = 1 | + 2-s − 4-s − 2·5-s − 7-s − 3·8-s − 3·9-s − 2·10-s + 11-s + 13-s − 14-s − 16-s − 2·17-s − 3·18-s + 2·20-s + 22-s − 25-s + 26-s + 28-s − 6·29-s + 4·31-s + 5·32-s − 2·34-s + 2·35-s + 3·36-s − 6·37-s + 6·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s − 1.06·8-s − 9-s − 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s + 1/2·36-s − 0.986·37-s + 0.948·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670558069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670558069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52547783052329, −12.02941586056368, −11.67188323896668, −11.48794393061791, −10.70793323742341, −10.44350315126702, −9.702173976016263, −9.171165140357985, −8.919412053020737, −8.443576780563135, −8.040589119850447, −7.406780502517677, −6.992831958784716, −6.241537932472678, −6.042789345028344, −5.381035633457802, −5.088045307865676, −4.308215244247740, −4.000136482581888, −3.540857249097217, −3.146275763714309, −2.477420294411381, −1.920797858062083, −0.7507343898666784, −0.4229661238286349,
0.4229661238286349, 0.7507343898666784, 1.920797858062083, 2.477420294411381, 3.146275763714309, 3.540857249097217, 4.000136482581888, 4.308215244247740, 5.088045307865676, 5.381035633457802, 6.042789345028344, 6.241537932472678, 6.992831958784716, 7.406780502517677, 8.040589119850447, 8.443576780563135, 8.919412053020737, 9.171165140357985, 9.702173976016263, 10.44350315126702, 10.70793323742341, 11.48794393061791, 11.67188323896668, 12.02941586056368, 12.52547783052329
