| L(s) = 1 | + 1.53·2-s + 1.17·3-s + 0.369·4-s − 5-s + 1.80·6-s + 7-s − 2.51·8-s − 1.63·9-s − 1.53·10-s + 0.431·12-s + 0.0917·13-s + 1.53·14-s − 1.17·15-s − 4.60·16-s + 5.51·17-s − 2.51·18-s − 0.921·19-s − 0.369·20-s + 1.17·21-s − 5.70·23-s − 2.93·24-s + 25-s + 0.141·26-s − 5.41·27-s + 0.369·28-s − 1.41·29-s − 1.80·30-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.675·3-s + 0.184·4-s − 0.447·5-s + 0.735·6-s + 0.377·7-s − 0.887·8-s − 0.543·9-s − 0.486·10-s + 0.124·12-s + 0.0254·13-s + 0.411·14-s − 0.302·15-s − 1.15·16-s + 1.33·17-s − 0.591·18-s − 0.211·19-s − 0.0825·20-s + 0.255·21-s − 1.19·23-s − 0.599·24-s + 0.200·25-s + 0.0276·26-s − 1.04·27-s + 0.0697·28-s − 0.263·29-s − 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 13 | \( 1 - 0.0917T + 13T^{2} \) |
| 17 | \( 1 - 5.51T + 17T^{2} \) |
| 19 | \( 1 + 0.921T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 0.879T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 + 5.90T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 - 3.03T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 + 2.83T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.133093016670968191923235482055, −7.37562977029193793833672257068, −6.33719938254107113860152692902, −5.64088101164823871091446039586, −5.00016759672718433075283058947, −4.11146135719352011891653453831, −3.45435998833542667408520947545, −2.87578162814776096265862687262, −1.73807617984567816411014982642, 0,
1.73807617984567816411014982642, 2.87578162814776096265862687262, 3.45435998833542667408520947545, 4.11146135719352011891653453831, 5.00016759672718433075283058947, 5.64088101164823871091446039586, 6.33719938254107113860152692902, 7.37562977029193793833672257068, 8.133093016670968191923235482055
