| L(s) = 1 | − 2.44·3-s + 5-s + 2.44·7-s + 2.99·9-s − 5.09·11-s − 4·13-s − 2.44·15-s − 5.24·17-s − 2.44·19-s − 5.99·21-s + 5.09·23-s + 25-s − 29-s + 7.34·31-s + 12.4·33-s + 2.44·35-s + 9.24·37-s + 9.79·39-s − 8·41-s + 7.74·43-s + 2.99·45-s + 12.6·47-s − 1.00·49-s + 12.8·51-s − 8·53-s − 5.09·55-s + 5.99·57-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.447·5-s + 0.925·7-s + 0.999·9-s − 1.53·11-s − 1.10·13-s − 0.632·15-s − 1.27·17-s − 0.561·19-s − 1.30·21-s + 1.06·23-s + 0.200·25-s − 0.185·29-s + 1.31·31-s + 2.17·33-s + 0.414·35-s + 1.51·37-s + 1.56·39-s − 1.24·41-s + 1.18·43-s + 0.447·45-s + 1.84·47-s − 0.142·49-s + 1.79·51-s − 1.09·53-s − 0.687·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 0.200T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + 0.755T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
| 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
−7.22311466766616228764419238803, −6.63621365063429023098490009854, −5.88086383548929323247525373989, −5.22707525050213482509854208040, −4.81292706774008372298469450681, −4.32016443962882764092418546688, −2.67525479405873706509349313835, −2.27764878538914105480884237245, −0.992115198879370387654978850270, 0,
0.992115198879370387654978850270, 2.27764878538914105480884237245, 2.67525479405873706509349313835, 4.32016443962882764092418546688, 4.81292706774008372298469450681, 5.22707525050213482509854208040, 5.88086383548929323247525373989, 6.63621365063429023098490009854, 7.22311466766616228764419238803
