| L(s) = 1 | − 3·2-s + 3.72·3-s + 4-s + 5·5-s − 11.1·6-s − 25.6·7-s + 21·8-s − 13.1·9-s − 15·10-s + 12.6·11-s + 3.72·12-s + 8.16·13-s + 76.8·14-s + 18.6·15-s − 71·16-s − 76.0·17-s + 39.4·18-s − 103.·19-s + 5·20-s − 95.2·21-s − 37.8·22-s − 23·23-s + 78.1·24-s + 25·25-s − 24.4·26-s − 149.·27-s − 25.6·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.715·3-s + 0.125·4-s + 0.447·5-s − 0.759·6-s − 1.38·7-s + 0.928·8-s − 0.487·9-s − 0.474·10-s + 0.345·11-s + 0.0894·12-s + 0.174·13-s + 1.46·14-s + 0.320·15-s − 1.10·16-s − 1.08·17-s + 0.516·18-s − 1.24·19-s + 0.0559·20-s − 0.989·21-s − 0.366·22-s − 0.208·23-s + 0.664·24-s + 0.200·25-s − 0.184·26-s − 1.06·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 3T + 8T^{2} \) |
| 3 | \( 1 - 3.72T + 27T^{2} \) |
| 7 | \( 1 + 25.6T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.16T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 63.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 239.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 2.31T + 1.48e5T^{2} \) |
| 59 | \( 1 - 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 294.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 286.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 551.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 21.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−12.90761089136512465265081162111, −11.15066028919431084835545334230, −10.07645973093231517023869918832, −9.123553392025433256140379167940, −8.712876977324397828325486558346, −7.25291281403127805344102946107, −6.02840033989751799276671915766, −3.93146318557273992403547348532, −2.25603207464750962996633996008, 0,
2.25603207464750962996633996008, 3.93146318557273992403547348532, 6.02840033989751799276671915766, 7.25291281403127805344102946107, 8.712876977324397828325486558346, 9.123553392025433256140379167940, 10.07645973093231517023869918832, 11.15066028919431084835545334230, 12.90761089136512465265081162111
