| L(s) = 1 | + 0.571·2-s + 1.62·3-s − 1.67·4-s + 0.928·6-s + 7-s − 2.09·8-s − 0.360·9-s + 3.19·11-s − 2.71·12-s − 4.39·13-s + 0.571·14-s + 2.14·16-s − 0.942·17-s − 0.206·18-s + 0.673·19-s + 1.62·21-s + 1.82·22-s − 23-s − 3.40·24-s − 2.50·26-s − 5.45·27-s − 1.67·28-s − 3.54·29-s − 0.497·31-s + 5.42·32-s + 5.18·33-s − 0.538·34-s + ⋯ |
| L(s) = 1 | + 0.404·2-s + 0.937·3-s − 0.836·4-s + 0.378·6-s + 0.377·7-s − 0.742·8-s − 0.120·9-s + 0.962·11-s − 0.784·12-s − 1.21·13-s + 0.152·14-s + 0.536·16-s − 0.228·17-s − 0.0485·18-s + 0.154·19-s + 0.354·21-s + 0.388·22-s − 0.208·23-s − 0.696·24-s − 0.492·26-s − 1.05·27-s − 0.316·28-s − 0.658·29-s − 0.0893·31-s + 0.959·32-s + 0.902·33-s − 0.0923·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 + 0.942T + 17T^{2} \) |
| 19 | \( 1 - 0.673T + 19T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 0.497T + 31T^{2} \) |
| 37 | \( 1 + 3.72T + 37T^{2} \) |
| 41 | \( 1 + 0.926T + 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 9.42T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 + 0.363T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 + 0.628T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
| 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
−8.212301524304928691931870763422, −7.50738270933617418725080275318, −6.64032816522458763266271067743, −5.65180167789574421478629798729, −4.97049537019726439503293203475, −4.17104034612615990824782118491, −3.49568972178021739859064496811, −2.66300927703831524457640514110, −1.61937015812938280906842112862, 0,
1.61937015812938280906842112862, 2.66300927703831524457640514110, 3.49568972178021739859064496811, 4.17104034612615990824782118491, 4.97049537019726439503293203475, 5.65180167789574421478629798729, 6.64032816522458763266271067743, 7.50738270933617418725080275318, 8.212301524304928691931870763422
