| L(s) = 1 | − 2.51·2-s + 2.61·3-s + 4.33·4-s − 6.59·6-s + 1.23·7-s − 5.88·8-s + 3.86·9-s − 2.94·11-s + 11.3·12-s − 3.09·14-s + 6.14·16-s − 5.86·17-s − 9.72·18-s − 4.76·19-s + 3.22·21-s + 7.42·22-s + 1.69·23-s − 15.4·24-s + 2.25·27-s + 5.33·28-s + 4.17·29-s − 2.50·31-s − 3.69·32-s − 7.72·33-s + 14.7·34-s + 16.7·36-s + 3.34·37-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 1.51·3-s + 2.16·4-s − 2.69·6-s + 0.465·7-s − 2.08·8-s + 1.28·9-s − 0.888·11-s + 3.28·12-s − 0.828·14-s + 1.53·16-s − 1.42·17-s − 2.29·18-s − 1.09·19-s + 0.703·21-s + 1.58·22-s + 0.353·23-s − 3.14·24-s + 0.434·27-s + 1.00·28-s + 0.774·29-s − 0.449·31-s − 0.652·32-s − 1.34·33-s + 2.53·34-s + 2.79·36-s + 0.549·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 8.95T + 41T^{2} \) |
| 43 | \( 1 + 0.795T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 0.0348T + 59T^{2} \) |
| 61 | \( 1 - 4.01T + 61T^{2} \) |
| 67 | \( 1 - 7.22T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.366805780150667741230191729154, −7.70638046946593393916271585562, −6.99794140325605447795145380126, −6.34454428439084551147280087133, −4.97573440658011413847082832830, −3.98100766264175903714381469334, −2.76493083254242426771630866979, −2.32648732790740781063133937429, −1.52040005389671110783951456353, 0,
1.52040005389671110783951456353, 2.32648732790740781063133937429, 2.76493083254242426771630866979, 3.98100766264175903714381469334, 4.97573440658011413847082832830, 6.34454428439084551147280087133, 6.99794140325605447795145380126, 7.70638046946593393916271585562, 8.366805780150667741230191729154
