| L(s) = 1 | − 505.·2-s + 1.31e4·3-s + 1.89e5·4-s − 6.63e6·6-s − 6.28e7·8-s + 1.29e8·9-s + 2.49e9·12-s + 2.27e8·13-s + 1.93e10·16-s − 6.52e10·18-s + 7.83e10·23-s − 8.24e11·24-s + 1.52e11·25-s − 1.14e11·26-s + 1.12e12·27-s − 7.17e11·29-s − 1.33e11·31-s − 5.64e12·32-s + 2.45e13·36-s + 2.98e12·39-s + 1.04e13·41-s − 3.95e13·46-s − 3.79e13·47-s + 2.53e14·48-s + 3.32e13·49-s − 7.71e13·50-s + 4.32e13·52-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 1.99·3-s + 2.89·4-s − 3.94·6-s − 3.74·8-s + 2.99·9-s + 5.79·12-s + 0.278·13-s + 4.50·16-s − 5.92·18-s + 23-s − 7.49·24-s + 25-s − 0.550·26-s + 3.99·27-s − 1.43·29-s − 0.156·31-s − 5.13·32-s + 8.69·36-s + 0.557·39-s + 1.30·41-s − 1.97·46-s − 1.59·47-s + 9.00·48-s + 49-s − 1.97·50-s + 0.808·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.978685878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978685878\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 7.83e10T \) |
| good | 2 | \( 1 + 505.T + 6.55e4T^{2} \) |
| 3 | \( 1 - 1.31e4T + 4.30e7T^{2} \) |
| 5 | \( 1 - 1.52e11T^{2} \) |
| 7 | \( 1 - 3.32e13T^{2} \) |
| 11 | \( 1 - 4.59e16T^{2} \) |
| 13 | \( 1 - 2.27e8T + 6.65e17T^{2} \) |
| 17 | \( 1 - 4.86e19T^{2} \) |
| 19 | \( 1 - 2.88e20T^{2} \) |
| 29 | \( 1 + 7.17e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + 1.33e11T + 7.27e23T^{2} \) |
| 37 | \( 1 - 1.23e25T^{2} \) |
| 41 | \( 1 - 1.04e13T + 6.37e25T^{2} \) |
| 43 | \( 1 - 1.36e26T^{2} \) |
| 47 | \( 1 + 3.79e13T + 5.66e26T^{2} \) |
| 53 | \( 1 - 3.87e27T^{2} \) |
| 59 | \( 1 + 6.02e13T + 2.15e28T^{2} \) |
| 61 | \( 1 - 3.67e28T^{2} \) |
| 67 | \( 1 - 1.64e29T^{2} \) |
| 71 | \( 1 - 5.79e14T + 4.16e29T^{2} \) |
| 73 | \( 1 - 8.85e14T + 6.50e29T^{2} \) |
| 79 | \( 1 - 2.30e30T^{2} \) |
| 83 | \( 1 - 5.07e30T^{2} \) |
| 89 | \( 1 - 1.54e31T^{2} \) |
| 97 | \( 1 - 6.14e31T^{2} \) |
| 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
−14.62858611386184425965840941705, −12.80286444578359497850164654342, −10.82803332061811798617184074373, −9.559937044721455388269512012472, −8.873758660945888926627681352902, −7.87689846616542435416416687270, −6.93079258941494336740045652652, −3.30237062375205593340606069317, −2.21367948598989338945496235546, −1.09900690167474916942541225891,
1.09900690167474916942541225891, 2.21367948598989338945496235546, 3.30237062375205593340606069317, 6.93079258941494336740045652652, 7.87689846616542435416416687270, 8.873758660945888926627681352902, 9.559937044721455388269512012472, 10.82803332061811798617184074373, 12.80286444578359497850164654342, 14.62858611386184425965840941705
