| L(s) = 1 | − 2.20·2-s + 2.48·3-s + 2.86·4-s − 1.68·5-s − 5.48·6-s + 0.336·7-s − 1.91·8-s + 3.17·9-s + 3.71·10-s + 11-s + 7.12·12-s − 0.741·14-s − 4.18·15-s − 1.51·16-s + 0.144·17-s − 7.00·18-s − 5.22·19-s − 4.82·20-s + 0.835·21-s − 2.20·22-s − 4.38·23-s − 4.75·24-s − 2.16·25-s + 0.435·27-s + 0.964·28-s − 2.44·29-s + 9.22·30-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.43·3-s + 1.43·4-s − 0.752·5-s − 2.23·6-s + 0.127·7-s − 0.676·8-s + 1.05·9-s + 1.17·10-s + 0.301·11-s + 2.05·12-s − 0.198·14-s − 1.08·15-s − 0.378·16-s + 0.0349·17-s − 1.65·18-s − 1.19·19-s − 1.07·20-s + 0.182·21-s − 0.470·22-s − 0.914·23-s − 0.970·24-s − 0.433·25-s + 0.0837·27-s + 0.182·28-s − 0.453·29-s + 1.68·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 - 0.336T + 7T^{2} \) |
| 17 | \( 1 - 0.144T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 + 0.887T + 47T^{2} \) |
| 53 | \( 1 + 5.33T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
| 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
−8.876935013403114407071007230818, −8.154032684672609904641708689579, −7.67626491961569215349145818032, −7.10254283392940963504620125452, −5.95033737419480637573012233042, −4.30465967749883273768961614495, −3.66356534827019679375860492083, −2.42161276756488061521686887684, −1.67539305102370448548239980460, 0,
1.67539305102370448548239980460, 2.42161276756488061521686887684, 3.66356534827019679375860492083, 4.30465967749883273768961614495, 5.95033737419480637573012233042, 7.10254283392940963504620125452, 7.67626491961569215349145818032, 8.154032684672609904641708689579, 8.876935013403114407071007230818
