| L(s) = 1 | + 2.04·2-s + 2.19·4-s + 5-s − 3.04·7-s + 0.405·8-s + 2.04·10-s + 1.89·11-s − 6.24·14-s − 3.56·16-s + 4.09·17-s − 2.58·19-s + 2.19·20-s + 3.87·22-s − 3.02·23-s + 25-s − 6.70·28-s − 3.30·29-s − 9.31·31-s − 8.11·32-s + 8.39·34-s − 3.04·35-s + 10.2·37-s − 5.29·38-s + 0.405·40-s + 0.731·41-s − 11.9·43-s + 4.15·44-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.09·4-s + 0.447·5-s − 1.15·7-s + 0.143·8-s + 0.647·10-s + 0.569·11-s − 1.66·14-s − 0.891·16-s + 0.993·17-s − 0.592·19-s + 0.491·20-s + 0.825·22-s − 0.631·23-s + 0.200·25-s − 1.26·28-s − 0.614·29-s − 1.67·31-s − 1.43·32-s + 1.43·34-s − 0.515·35-s + 1.69·37-s − 0.858·38-s + 0.0641·40-s + 0.114·41-s − 1.82·43-s + 0.626·44-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 | \( 1 - 2.04T + 2T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 3.30T + 29T^{2} \) |
| 31 | \( 1 + 9.31T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 0.731T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 8.98T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.71T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 - 7.59T + 97T^{2} \) |
| 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
−7.18961190565451005710431813538, −6.48343026125418053638686984476, −6.07009049256669712473649726337, −5.47777530410111842797644926547, −4.69282628753967361673502424330, −3.78092358056871476818721158232, −3.42630288301693866463196223153, −2.57290533550262596138727757840, −1.61892923925009960658245018152, 0,
1.61892923925009960658245018152, 2.57290533550262596138727757840, 3.42630288301693866463196223153, 3.78092358056871476818721158232, 4.69282628753967361673502424330, 5.47777530410111842797644926547, 6.07009049256669712473649726337, 6.48343026125418053638686984476, 7.18961190565451005710431813538
