| L(s) = 1 | − 2.47·2-s + 4.12·4-s − 0.973·7-s − 5.25·8-s + 5.38·11-s − 1.99·13-s + 2.40·14-s + 4.74·16-s + 2.04·17-s + 6.20·19-s − 13.3·22-s − 1.93·23-s + 4.94·26-s − 4.01·28-s − 4.81·29-s − 6.64·31-s − 1.24·32-s − 5.05·34-s − 0.978·37-s − 15.3·38-s − 2.73·41-s + 3.99·43-s + 22.1·44-s + 4.79·46-s − 7.21·47-s − 6.05·49-s − 8.23·52-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.06·4-s − 0.367·7-s − 1.85·8-s + 1.62·11-s − 0.553·13-s + 0.643·14-s + 1.18·16-s + 0.495·17-s + 1.42·19-s − 2.83·22-s − 0.404·23-s + 0.968·26-s − 0.758·28-s − 0.894·29-s − 1.19·31-s − 0.220·32-s − 0.866·34-s − 0.160·37-s − 2.49·38-s − 0.426·41-s + 0.609·43-s + 3.34·44-s + 0.707·46-s − 1.05·47-s − 0.864·49-s − 1.14·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 7 | \( 1 + 0.973T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 3.99T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
| 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
−7.76217454619415305068543904974, −7.33959352553552302825154606785, −6.63191761839772602187846746582, −6.01074621225195197813571304519, −5.01461576198382520906998662961, −3.77168929719854540336354769298, −3.07860339992923894236025109129, −1.85499624633936524675312773615, −1.22658844341773594422014445228, 0,
1.22658844341773594422014445228, 1.85499624633936524675312773615, 3.07860339992923894236025109129, 3.77168929719854540336354769298, 5.01461576198382520906998662961, 6.01074621225195197813571304519, 6.63191761839772602187846746582, 7.33959352553552302825154606785, 7.76217454619415305068543904974
