| L(s) = 1 | − 2.09·2-s − 1.47·3-s + 2.39·4-s − 0.522·5-s + 3.09·6-s − 0.294·7-s − 0.817·8-s − 0.817·9-s + 1.09·10-s + 2.41·11-s − 3.53·12-s − 0.845·13-s + 0.618·14-s + 0.772·15-s − 3.06·16-s + 7.69·17-s + 1.71·18-s − 1.02·19-s − 1.24·20-s + 0.435·21-s − 5.06·22-s + 0.353·23-s + 1.20·24-s − 4.72·25-s + 1.77·26-s + 5.63·27-s − 0.705·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.852·3-s + 1.19·4-s − 0.233·5-s + 1.26·6-s − 0.111·7-s − 0.289·8-s − 0.272·9-s + 0.346·10-s + 0.729·11-s − 1.01·12-s − 0.234·13-s + 0.165·14-s + 0.199·15-s − 0.766·16-s + 1.86·17-s + 0.403·18-s − 0.235·19-s − 0.279·20-s + 0.0950·21-s − 1.08·22-s + 0.0737·23-s + 0.246·24-s − 0.945·25-s + 0.347·26-s + 1.08·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 431 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 + 0.522T + 5T^{2} \) |
| 7 | \( 1 + 0.294T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 0.845T + 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 - 0.353T + 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 - 3.54T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 - 9.20T + 97T^{2} \) |
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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
−10.49741387678669712748664808452, −9.850292743137658648520149616898, −8.975904529257578709805168778784, −8.020210797461077444404711404657, −7.21911881506331381433316710478, −6.17237951805918348387135627855, −5.14488017911330079642349793194, −3.51790462767506543012504496676, −1.57497582503403003640892917893, 0,
1.57497582503403003640892917893, 3.51790462767506543012504496676, 5.14488017911330079642349793194, 6.17237951805918348387135627855, 7.21911881506331381433316710478, 8.020210797461077444404711404657, 8.975904529257578709805168778784, 9.850292743137658648520149616898, 10.49741387678669712748664808452
