| L(s) = 1 | + 2.30·2-s + 3-s + 3.30·4-s + 5-s + 2.30·6-s + 7-s + 3.00·8-s − 2·9-s + 2.30·10-s − 1.60·11-s + 3.30·12-s + 2.30·14-s + 15-s + 0.302·16-s + 7.60·17-s − 4.60·18-s + 5.60·19-s + 3.30·20-s + 21-s − 3.69·22-s − 3·23-s + 3.00·24-s + 25-s − 5·27-s + 3.30·28-s − 6.21·29-s + 2.30·30-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.447·5-s + 0.940·6-s + 0.377·7-s + 1.06·8-s − 0.666·9-s + 0.728·10-s − 0.484·11-s + 0.953·12-s + 0.615·14-s + 0.258·15-s + 0.0756·16-s + 1.84·17-s − 1.08·18-s + 1.28·19-s + 0.738·20-s + 0.218·21-s − 0.788·22-s − 0.625·23-s + 0.612·24-s + 0.200·25-s − 0.962·27-s + 0.624·28-s − 1.15·29-s + 0.420·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.741236432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.741236432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 0.788T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 6.21T + 89T^{2} \) |
| 97 | \( 1 - 8.39T + 97T^{2} \) |
| 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
−10.23948127815180831698568867260, −9.491597028274481016595490014246, −8.249182346494674831196633259830, −7.58170703178011570551686526041, −6.39699950830088178096986400096, −5.41123508753474257536257669286, −5.11624718349095016440146693414, −3.56770180873369175370136759490, −3.08475022846135840260877917843, −1.86263334625763701777694191280,
1.86263334625763701777694191280, 3.08475022846135840260877917843, 3.56770180873369175370136759490, 5.11624718349095016440146693414, 5.41123508753474257536257669286, 6.39699950830088178096986400096, 7.58170703178011570551686526041, 8.249182346494674831196633259830, 9.491597028274481016595490014246, 10.23948127815180831698568867260
