| L(s) = 1 | − 4·2-s − 18.7·3-s + 16·4-s + 2.23·5-s + 74.9·6-s − 36.2·7-s − 64·8-s + 107.·9-s − 8.95·10-s − 24.0·11-s − 299.·12-s + 160.·13-s + 145.·14-s − 41.9·15-s + 256·16-s + 2.08e3·17-s − 430.·18-s + 1.52e3·19-s + 35.8·20-s + 679.·21-s + 96.3·22-s − 272.·23-s + 1.19e3·24-s − 3.11e3·25-s − 641.·26-s + 2.53e3·27-s − 580.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.20·3-s + 0.5·4-s + 0.0400·5-s + 0.849·6-s − 0.279·7-s − 0.353·8-s + 0.443·9-s − 0.0283·10-s − 0.0600·11-s − 0.600·12-s + 0.263·13-s + 0.197·14-s − 0.0481·15-s + 0.250·16-s + 1.74·17-s − 0.313·18-s + 0.967·19-s + 0.0200·20-s + 0.336·21-s + 0.0424·22-s − 0.107·23-s + 0.424·24-s − 0.998·25-s − 0.186·26-s + 0.668·27-s − 0.139·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 79 | \( 1 + 6.24e3T \) |
| good | 3 | \( 1 + 18.7T + 243T^{2} \) |
| 5 | \( 1 - 2.23T + 3.12e3T^{2} \) |
| 7 | \( 1 + 36.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 24.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 160.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 272.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.90e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.42e4T + 2.07e9T^{2} \) |
| 83 | \( 1 + 4.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.02e4T + 8.58e9T^{2} \) |
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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
−11.71100422756637694229631417441, −10.34522384068228819423090078957, −9.794786421426255951628933439511, −8.346673099053611122374183374923, −7.22317103977660775941773021532, −6.05994512360458229083820325020, −5.24470798679363576196825132691, −3.32603148033370715846445017822, −1.31111086363410367411504186081, 0,
1.31111086363410367411504186081, 3.32603148033370715846445017822, 5.24470798679363576196825132691, 6.05994512360458229083820325020, 7.22317103977660775941773021532, 8.346673099053611122374183374923, 9.794786421426255951628933439511, 10.34522384068228819423090078957, 11.71100422756637694229631417441
