| L(s) = 1 | + 5.05·2-s + 17.5·4-s − 12.9·5-s − 7·7-s + 48.4·8-s − 65.4·10-s − 11·11-s − 55.5·13-s − 35.3·14-s + 104.·16-s − 59.2·17-s − 33.1·19-s − 227.·20-s − 55.6·22-s − 26.0·23-s + 42.5·25-s − 280.·26-s − 123.·28-s + 188.·29-s − 278.·31-s + 139.·32-s − 299.·34-s + 90.5·35-s + 201.·37-s − 167.·38-s − 626.·40-s + 126.·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.19·4-s − 1.15·5-s − 0.377·7-s + 2.13·8-s − 2.06·10-s − 0.301·11-s − 1.18·13-s − 0.675·14-s + 1.62·16-s − 0.845·17-s − 0.400·19-s − 2.54·20-s − 0.539·22-s − 0.236·23-s + 0.340·25-s − 2.11·26-s − 0.830·28-s + 1.20·29-s − 1.61·31-s + 0.772·32-s − 1.51·34-s + 0.437·35-s + 0.895·37-s − 0.716·38-s − 2.47·40-s + 0.482·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.05T + 8T^{2} \) |
| 5 | \( 1 + 12.9T + 125T^{2} \) |
| 13 | \( 1 + 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 79.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 49.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 809.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 85.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 451.T + 9.12e5T^{2} \) |
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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
−9.914581933846128645694264623392, −8.531221814888518887813241355634, −7.43908560853462589685585361744, −6.88384612999257518466981169394, −5.83209613863528777573237939348, −4.77636070803062740776977182768, −4.17270610490476072193330805663, −3.20097051193696083071281728439, −2.23792866836422627527643145512, 0,
2.23792866836422627527643145512, 3.20097051193696083071281728439, 4.17270610490476072193330805663, 4.77636070803062740776977182768, 5.83209613863528777573237939348, 6.88384612999257518466981169394, 7.43908560853462589685585361744, 8.531221814888518887813241355634, 9.914581933846128645694264623392
