| L(s) = 1 | + 1.90·2-s − 3·3-s − 4.36·4-s − 5.71·6-s + 22.9·7-s − 23.5·8-s + 9·9-s + 11·11-s + 13.0·12-s − 66.7·13-s + 43.6·14-s − 10.0·16-s + 3.45·17-s + 17.1·18-s + 78.2·19-s − 68.7·21-s + 20.9·22-s − 12.2·23-s + 70.7·24-s − 127.·26-s − 27·27-s − 100.·28-s − 31.1·29-s + 247.·31-s + 169.·32-s − 33·33-s + 6.58·34-s + ⋯ |
| L(s) = 1 | + 0.674·2-s − 0.577·3-s − 0.545·4-s − 0.389·6-s + 1.23·7-s − 1.04·8-s + 0.333·9-s + 0.301·11-s + 0.315·12-s − 1.42·13-s + 0.834·14-s − 0.156·16-s + 0.0493·17-s + 0.224·18-s + 0.944·19-s − 0.714·21-s + 0.203·22-s − 0.111·23-s + 0.601·24-s − 0.959·26-s − 0.192·27-s − 0.675·28-s − 0.199·29-s + 1.43·31-s + 0.936·32-s − 0.174·33-s + 0.0332·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.90T + 8T^{2} \) |
| 7 | \( 1 - 22.9T + 343T^{2} \) |
| 13 | \( 1 + 66.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.45T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 12.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 304.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 29.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 269.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 16.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 28.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 853.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 36.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 23.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 707.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 412.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.545219535754985454645743513721, −8.507383788613095679588096596369, −7.66971650226535703084286455320, −6.67428184122129793160850072834, −5.48066244628059150307027467943, −4.95270927244098311351284433210, −4.27893732612711307807493969845, −2.96090904958045158740593924008, −1.46064607255351147116592512346, 0,
1.46064607255351147116592512346, 2.96090904958045158740593924008, 4.27893732612711307807493969845, 4.95270927244098311351284433210, 5.48066244628059150307027467943, 6.67428184122129793160850072834, 7.66971650226535703084286455320, 8.507383788613095679588096596369, 9.545219535754985454645743513721
