| L(s) = 1 | − 0.923·3-s + 0.670·5-s + 28.6·7-s − 26.1·9-s + 24.1·11-s + 17.5·13-s − 0.619·15-s − 61.1·17-s − 19·19-s − 26.4·21-s − 160.·23-s − 124.·25-s + 49.0·27-s − 287.·29-s + 47.5·31-s − 22.2·33-s + 19.2·35-s + 237.·37-s − 16.1·39-s + 197.·41-s + 352.·43-s − 17.5·45-s − 357.·47-s + 479.·49-s + 56.4·51-s − 381.·53-s + 16.1·55-s + ⋯ |
| L(s) = 1 | − 0.177·3-s + 0.0600·5-s + 1.54·7-s − 0.968·9-s + 0.661·11-s + 0.374·13-s − 0.0106·15-s − 0.872·17-s − 0.229·19-s − 0.275·21-s − 1.45·23-s − 0.996·25-s + 0.349·27-s − 1.84·29-s + 0.275·31-s − 0.117·33-s + 0.0929·35-s + 1.05·37-s − 0.0664·39-s + 0.753·41-s + 1.25·43-s − 0.0581·45-s − 1.10·47-s + 1.39·49-s + 0.155·51-s − 0.988·53-s + 0.0396·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 0.923T + 27T^{2} \) |
| 5 | \( 1 - 0.670T + 125T^{2} \) |
| 7 | \( 1 - 28.6T + 343T^{2} \) |
| 11 | \( 1 - 24.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 381.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 827.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 296.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 916.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 547.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 257.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 206.T + 9.12e5T^{2} \) |
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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
−8.859771476675852534700498913300, −8.128618289936864902701336090800, −7.52081812368166376830493379982, −6.18457559899453827850369349926, −5.70586300060151477093549077800, −4.57234708248432944383513572191, −3.87539066830211420757314647818, −2.35630011290541852603678874675, −1.51106345090057147817616393304, 0,
1.51106345090057147817616393304, 2.35630011290541852603678874675, 3.87539066830211420757314647818, 4.57234708248432944383513572191, 5.70586300060151477093549077800, 6.18457559899453827850369349926, 7.52081812368166376830493379982, 8.128618289936864902701336090800, 8.859771476675852534700498913300
