| L(s) = 1 | − 2.63·2-s + 3.04·3-s + 4.91·4-s − 5-s − 8.00·6-s − 0.574·7-s − 7.67·8-s + 6.26·9-s + 2.63·10-s + 2.57·11-s + 14.9·12-s − 0.468·13-s + 1.51·14-s − 3.04·15-s + 10.3·16-s − 4.08·17-s − 16.4·18-s + 19-s − 4.91·20-s − 1.74·21-s − 6.77·22-s + 1.51·23-s − 23.3·24-s + 25-s + 1.23·26-s + 9.92·27-s − 2.82·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.75·3-s + 2.45·4-s − 0.447·5-s − 3.26·6-s − 0.217·7-s − 2.71·8-s + 2.08·9-s + 0.831·10-s + 0.776·11-s + 4.31·12-s − 0.129·13-s + 0.403·14-s − 0.785·15-s + 2.58·16-s − 0.991·17-s − 3.88·18-s + 0.229·19-s − 1.09·20-s − 0.381·21-s − 1.44·22-s + 0.315·23-s − 4.76·24-s + 0.200·25-s + 0.241·26-s + 1.90·27-s − 0.534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7270491014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7270491014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 0.468T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.574T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
| 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
−14.36700061896243089115208569393, −12.91304343132066980748670025179, −11.49416628978873468021726546887, −10.27405079109865726173398810264, −9.105757195727031777406727113277, −8.840094616842938591313698357814, −7.62868382943286804205914427415, −6.89000884209947823183860317021, −3.53759944795802409143405612967, −1.99194710362536732940468895900,
1.99194710362536732940468895900, 3.53759944795802409143405612967, 6.89000884209947823183860317021, 7.62868382943286804205914427415, 8.840094616842938591313698357814, 9.105757195727031777406727113277, 10.27405079109865726173398810264, 11.49416628978873468021726546887, 12.91304343132066980748670025179, 14.36700061896243089115208569393
