| L(s) = 1 | − 0.561·5-s + 0.561·7-s + 2·11-s − 13-s + 0.561·17-s − 6·19-s − 4.68·25-s + 8.24·29-s − 7.12·31-s − 0.315·35-s − 9.68·37-s − 7.12·41-s + 8.80·43-s + 1.68·47-s − 6.68·49-s + 4.87·53-s − 1.12·55-s − 6·59-s + 13.3·61-s + 0.561·65-s − 6·67-s − 1.68·71-s + 10·73-s + 1.12·77-s − 12·79-s − 17.3·83-s − 0.315·85-s + ⋯ |
| L(s) = 1 | − 0.251·5-s + 0.212·7-s + 0.603·11-s − 0.277·13-s + 0.136·17-s − 1.37·19-s − 0.936·25-s + 1.53·29-s − 1.27·31-s − 0.0533·35-s − 1.59·37-s − 1.11·41-s + 1.34·43-s + 0.245·47-s − 0.954·49-s + 0.669·53-s − 0.151·55-s − 0.781·59-s + 1.71·61-s + 0.0696·65-s − 0.733·67-s − 0.199·71-s + 1.17·73-s + 0.127·77-s − 1.35·79-s − 1.90·83-s − 0.0342·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.310080589934033374812375342089, −7.33713268528036076349461049532, −6.74069179733242319110287892029, −5.94728239294354965613847389707, −5.08379743035954149505590828294, −4.23642977810300404332541306626, −3.58569493556401252252580430744, −2.43470077497720609517368412384, −1.49264770548250293594637876598, 0,
1.49264770548250293594637876598, 2.43470077497720609517368412384, 3.58569493556401252252580430744, 4.23642977810300404332541306626, 5.08379743035954149505590828294, 5.94728239294354965613847389707, 6.74069179733242319110287892029, 7.33713268528036076349461049532, 8.310080589934033374812375342089
