| L(s) = 1 | − 2.43·2-s + 2.39·3-s + 3.92·4-s + 2.97·5-s − 5.82·6-s + 3.62·7-s − 4.69·8-s + 2.72·9-s − 7.24·10-s + 3.86·11-s + 9.40·12-s − 2.18·13-s − 8.82·14-s + 7.11·15-s + 3.58·16-s − 5.33·17-s − 6.63·18-s + 6.97·19-s + 11.6·20-s + 8.67·21-s − 9.41·22-s + 4.02·23-s − 11.2·24-s + 3.85·25-s + 5.32·26-s − 0.658·27-s + 14.2·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.38·3-s + 1.96·4-s + 1.33·5-s − 2.37·6-s + 1.37·7-s − 1.66·8-s + 0.908·9-s − 2.29·10-s + 1.16·11-s + 2.71·12-s − 0.606·13-s − 2.35·14-s + 1.83·15-s + 0.895·16-s − 1.29·17-s − 1.56·18-s + 1.60·19-s + 2.61·20-s + 1.89·21-s − 2.00·22-s + 0.839·23-s − 2.29·24-s + 0.770·25-s + 1.04·26-s − 0.126·27-s + 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.469501768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469501768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 - 5.84T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 73 | \( 1 - 4.80T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 - 5.81T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
| 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
−8.618957206490993572347000957934, −7.67332631245650637392931469973, −7.20707151910258812306352558837, −6.50388530444189847777507614986, −5.41710742997719041708416100601, −4.53757883868343689638895093682, −3.24219892942300094590996853736, −2.29456505580128056624565796448, −1.82444715516759643378402503465, −1.15431278364685595238280730402,
1.15431278364685595238280730402, 1.82444715516759643378402503465, 2.29456505580128056624565796448, 3.24219892942300094590996853736, 4.53757883868343689638895093682, 5.41710742997719041708416100601, 6.50388530444189847777507614986, 7.20707151910258812306352558837, 7.67332631245650637392931469973, 8.618957206490993572347000957934
