| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 2·5-s + 4.41·8-s + 4.82·10-s − 11-s − 0.828·13-s + 2.99·16-s + 4.41·17-s + 7.24·19-s + 7.65·20-s − 2.41·22-s + 7·23-s − 25-s − 1.99·26-s − 3.24·29-s + 5.65·31-s − 1.58·32-s + 10.6·34-s − 9.48·37-s + 17.4·38-s + 8.82·40-s + 1.17·41-s − 2.75·43-s − 3.82·44-s + 16.8·46-s − 9.82·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.894·5-s + 1.56·8-s + 1.52·10-s − 0.301·11-s − 0.229·13-s + 0.749·16-s + 1.07·17-s + 1.66·19-s + 1.71·20-s − 0.514·22-s + 1.45·23-s − 0.200·25-s − 0.392·26-s − 0.602·29-s + 1.01·31-s − 0.280·32-s + 1.82·34-s − 1.55·37-s + 2.83·38-s + 1.39·40-s + 0.182·41-s − 0.420·43-s − 0.577·44-s + 2.49·46-s − 1.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.050849270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.050849270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 - 7T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 + 9.82T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
| 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
−7.987364910507144321020014733148, −7.22062686295560240153687977722, −6.63016547074526999917319745606, −5.76757748163586321867570624091, −5.23561173656975590429736390450, −4.89790082122465375778514851087, −3.61599654611592607405100288231, −3.14110669469976902497524644594, −2.27893739810726249470998704493, −1.24970897769674826198021931009,
1.24970897769674826198021931009, 2.27893739810726249470998704493, 3.14110669469976902497524644594, 3.61599654611592607405100288231, 4.89790082122465375778514851087, 5.23561173656975590429736390450, 5.76757748163586321867570624091, 6.63016547074526999917319745606, 7.22062686295560240153687977722, 7.987364910507144321020014733148
