| L(s) = 1 | − 2.54·2-s + 2.90·3-s + 4.45·4-s − 2.39·5-s − 7.38·6-s + 0.864·7-s − 6.23·8-s + 5.44·9-s + 6.09·10-s + 0.754·11-s + 12.9·12-s + 6.48·13-s − 2.19·14-s − 6.97·15-s + 6.93·16-s − 2.24·17-s − 13.8·18-s + 4.96·19-s − 10.6·20-s + 2.51·21-s − 1.91·22-s − 5.99·23-s − 18.1·24-s + 0.752·25-s − 16.4·26-s + 7.11·27-s + 3.85·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 1.67·3-s + 2.22·4-s − 1.07·5-s − 3.01·6-s + 0.326·7-s − 2.20·8-s + 1.81·9-s + 1.92·10-s + 0.227·11-s + 3.73·12-s + 1.79·13-s − 0.587·14-s − 1.80·15-s + 1.73·16-s − 0.543·17-s − 3.26·18-s + 1.13·19-s − 2.38·20-s + 0.548·21-s − 0.408·22-s − 1.25·23-s − 3.69·24-s + 0.150·25-s − 3.23·26-s + 1.36·27-s + 0.728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 - 0.864T + 7T^{2} \) |
| 11 | \( 1 - 0.754T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + 8.49T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 8.33T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 4.88T + 97T^{2} \) |
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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
−7.82542825510079338349110435517, −7.33554489523347310326106139017, −6.64535481329727412249921533524, −5.65474150726245456083062479646, −4.12983799365862456779728585328, −3.64674686409070830809628273552, −2.98996289189027224652204057939, −1.80926101488528042198692119413, −1.46473868303422878830511208558, 0,
1.46473868303422878830511208558, 1.80926101488528042198692119413, 2.98996289189027224652204057939, 3.64674686409070830809628273552, 4.12983799365862456779728585328, 5.65474150726245456083062479646, 6.64535481329727412249921533524, 7.33554489523347310326106139017, 7.82542825510079338349110435517
