| L(s) = 1 | − 2-s + 0.180·3-s + 4-s − 3.11·5-s − 0.180·6-s + 2.83·7-s − 8-s − 2.96·9-s + 3.11·10-s + 1.84·11-s + 0.180·12-s − 3.19·13-s − 2.83·14-s − 0.561·15-s + 16-s + 5.26·17-s + 2.96·18-s − 2.30·19-s − 3.11·20-s + 0.510·21-s − 1.84·22-s + 23-s − 0.180·24-s + 4.71·25-s + 3.19·26-s − 1.07·27-s + 2.83·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.103·3-s + 0.5·4-s − 1.39·5-s − 0.0735·6-s + 1.07·7-s − 0.353·8-s − 0.989·9-s + 0.985·10-s + 0.555·11-s + 0.0519·12-s − 0.887·13-s − 0.758·14-s − 0.144·15-s + 0.250·16-s + 1.27·17-s + 0.699·18-s − 0.529·19-s − 0.696·20-s + 0.111·21-s − 0.392·22-s + 0.208·23-s − 0.0367·24-s + 0.942·25-s + 0.627·26-s − 0.206·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.180T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 - 2.83T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 5.26T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.161T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 + 5.41T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 0.272T + 97T^{2} \) |
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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
−7.81164111151874966877908445004, −7.47270357328666368683671747028, −6.46391934268394622184156136782, −5.56992246937627775351162803536, −4.79953379892272203046334183195, −3.96969523970249728078718261392, −3.18747850475691269886225674568, −2.26456493095544430721027210430, −1.11104974692636494669543908376, 0,
1.11104974692636494669543908376, 2.26456493095544430721027210430, 3.18747850475691269886225674568, 3.96969523970249728078718261392, 4.79953379892272203046334183195, 5.56992246937627775351162803536, 6.46391934268394622184156136782, 7.47270357328666368683671747028, 7.81164111151874966877908445004
