| L(s) = 1 | − 3-s − 2·4-s − 5-s − 5·7-s + 9-s + 2·12-s − 4·13-s + 15-s + 4·16-s + 3·17-s − 7·19-s + 2·20-s + 5·21-s − 23-s + 25-s − 27-s + 10·28-s − 29-s − 8·31-s + 5·35-s − 2·36-s + 4·37-s + 4·39-s + 2·41-s + 8·43-s − 45-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 0.727·17-s − 1.60·19-s + 0.447·20-s + 1.09·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.88·28-s − 0.185·29-s − 1.43·31-s + 0.845·35-s − 1/3·36-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.96723896522345, −16.56025374118693, −15.80740538187951, −15.31360521911730, −14.56010980869474, −14.14518773003852, −13.13051550392778, −12.87951994910884, −12.39131854562031, −12.05475819285217, −10.83727095593822, −10.53077059782511, −9.649119436232221, −9.474251251994725, −8.801455112179655, −7.851313481328604, −7.326852644115285, −6.541389317380707, −5.943229245718141, −5.336298575707143, −4.423323678366348, −3.903632830734584, −3.220621302043720, −2.263290874296234, −0.6742042071137102, 0,
0.6742042071137102, 2.263290874296234, 3.220621302043720, 3.903632830734584, 4.423323678366348, 5.336298575707143, 5.943229245718141, 6.541389317380707, 7.326852644115285, 7.851313481328604, 8.801455112179655, 9.474251251994725, 9.649119436232221, 10.53077059782511, 10.83727095593822, 12.05475819285217, 12.39131854562031, 12.87951994910884, 13.13051550392778, 14.14518773003852, 14.56010980869474, 15.31360521911730, 15.80740538187951, 16.56025374118693, 16.96723896522345
