| L(s) = 1 | − 1.80·2-s + 1.24·4-s + 5-s + 0.801·7-s + 1.35·8-s − 1.80·10-s − 11-s − 1.44·14-s − 4.93·16-s + 0.396·17-s − 2.74·19-s + 1.24·20-s + 1.80·22-s − 1.55·23-s + 25-s + 1.00·28-s + 7.34·29-s + 2.38·31-s + 6.18·32-s − 0.713·34-s + 0.801·35-s − 5.96·37-s + 4.93·38-s + 1.35·40-s − 8.80·41-s − 6.80·43-s − 1.24·44-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.623·4-s + 0.447·5-s + 0.303·7-s + 0.479·8-s − 0.569·10-s − 0.301·11-s − 0.386·14-s − 1.23·16-s + 0.0960·17-s − 0.628·19-s + 0.278·20-s + 0.384·22-s − 0.324·23-s + 0.200·25-s + 0.188·28-s + 1.36·29-s + 0.428·31-s + 1.09·32-s − 0.122·34-s + 0.135·35-s − 0.979·37-s + 0.801·38-s + 0.214·40-s − 1.37·41-s − 1.03·43-s − 0.187·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 - 0.801T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 - 0.396T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 + 8.80T + 41T^{2} \) |
| 43 | \( 1 + 6.80T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.868T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 - 9.93T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75898458555288602818189753293, −6.90875668307938392582431820275, −6.47539254565936839592891254847, −5.40999574431198587675399184427, −4.79985938722232849010931628765, −3.95075310271566759764650975042, −2.79425238308548319785102952387, −1.96352906260938269054984911867, −1.17012602857637456355746939806, 0,
1.17012602857637456355746939806, 1.96352906260938269054984911867, 2.79425238308548319785102952387, 3.95075310271566759764650975042, 4.79985938722232849010931628765, 5.40999574431198587675399184427, 6.47539254565936839592891254847, 6.90875668307938392582431820275, 7.75898458555288602818189753293
