| L(s) = 1 | + 2.55·2-s + 3-s + 4.51·4-s − 3.99·5-s + 2.55·6-s − 2.72·7-s + 6.40·8-s + 9-s − 10.2·10-s − 2.41·11-s + 4.51·12-s − 1.70·13-s − 6.96·14-s − 3.99·15-s + 7.33·16-s − 0.933·17-s + 2.55·18-s − 2.19·19-s − 18.0·20-s − 2.72·21-s − 6.17·22-s − 6.25·23-s + 6.40·24-s + 10.9·25-s − 4.35·26-s + 27-s − 12.3·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.577·3-s + 2.25·4-s − 1.78·5-s + 1.04·6-s − 1.03·7-s + 2.26·8-s + 0.333·9-s − 3.22·10-s − 0.729·11-s + 1.30·12-s − 0.472·13-s − 1.86·14-s − 1.03·15-s + 1.83·16-s − 0.226·17-s + 0.601·18-s − 0.504·19-s − 4.03·20-s − 0.595·21-s − 1.31·22-s − 1.30·23-s + 1.30·24-s + 2.19·25-s − 0.853·26-s + 0.192·27-s − 2.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 3.99T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 + 0.933T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 - 0.762T + 29T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 8.41T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.40T + 97T^{2} \) |
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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
−8.024328636449513930386220361306, −7.48624053343362112956095932568, −6.80823604477224271255965677221, −6.05886432784592398282340151727, −4.95400709254179942689723980517, −4.30905657773059188040656981804, −3.61095136434379374111813707735, −3.11586191143964506124838527781, −2.21412437677741036090903813237, 0,
2.21412437677741036090903813237, 3.11586191143964506124838527781, 3.61095136434379374111813707735, 4.30905657773059188040656981804, 4.95400709254179942689723980517, 6.05886432784592398282340151727, 6.80823604477224271255965677221, 7.48624053343362112956095932568, 8.024328636449513930386220361306
