| L(s) = 1 | − 0.0898·2-s − 1.99·4-s − 4.36·7-s + 0.358·8-s − 4.39·11-s + 1.98·13-s + 0.391·14-s + 3.95·16-s + 0.997·17-s + 1.35·19-s + 0.394·22-s + 2.35·23-s − 0.177·26-s + 8.68·28-s + 7.97·29-s − 3.67·31-s − 1.07·32-s − 0.0895·34-s + 1.43·37-s − 0.121·38-s + 5.98·41-s − 2.68·43-s + 8.74·44-s − 0.211·46-s + 10.9·47-s + 12.0·49-s − 3.94·52-s + ⋯ |
| L(s) = 1 | − 0.0635·2-s − 0.995·4-s − 1.64·7-s + 0.126·8-s − 1.32·11-s + 0.549·13-s + 0.104·14-s + 0.987·16-s + 0.241·17-s + 0.309·19-s + 0.0840·22-s + 0.491·23-s − 0.0349·26-s + 1.64·28-s + 1.48·29-s − 0.660·31-s − 0.189·32-s − 0.0153·34-s + 0.236·37-s − 0.0196·38-s + 0.934·41-s − 0.409·43-s + 1.31·44-s − 0.0312·46-s + 1.59·47-s + 1.71·49-s − 0.547·52-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 | \( 1 + 0.0898T + 2T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 0.997T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 - 5.98T + 41T^{2} \) |
| 43 | \( 1 + 2.68T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 - 0.424T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 - 0.737T + 83T^{2} \) |
| 89 | \( 1 - 9.78T + 89T^{2} \) |
| 97 | \( 1 + 0.0337T + 97T^{2} \) |
| 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
−7.73580459512069251613143667350, −7.21594760610295307167842482469, −6.15283607738235528889820329215, −5.72205134257058703351082241781, −4.86739846778158628865971941256, −4.04598592484486366911104080081, −3.20504966079342169783581106854, −2.67298803981766120882725320108, −0.988175731362714418909239816920, 0,
0.988175731362714418909239816920, 2.67298803981766120882725320108, 3.20504966079342169783581106854, 4.04598592484486366911104080081, 4.86739846778158628865971941256, 5.72205134257058703351082241781, 6.15283607738235528889820329215, 7.21594760610295307167842482469, 7.73580459512069251613143667350
