| L(s) = 1 | + 1.25·2-s + 3·3-s − 6.43·4-s − 5.99·5-s + 3.75·6-s − 7·7-s − 18.0·8-s + 9·9-s − 7.50·10-s + 2.45·11-s − 19.2·12-s − 8.39·13-s − 8.76·14-s − 17.9·15-s + 28.8·16-s + 41.7·17-s + 11.2·18-s + 87.5·19-s + 38.5·20-s − 21·21-s + 3.07·22-s + 23·23-s − 54.2·24-s − 89.0·25-s − 10.5·26-s + 27·27-s + 45.0·28-s + ⋯ |
| L(s) = 1 | + 0.442·2-s + 0.577·3-s − 0.804·4-s − 0.536·5-s + 0.255·6-s − 0.377·7-s − 0.798·8-s + 0.333·9-s − 0.237·10-s + 0.0673·11-s − 0.464·12-s − 0.179·13-s − 0.167·14-s − 0.309·15-s + 0.450·16-s + 0.596·17-s + 0.147·18-s + 1.05·19-s + 0.431·20-s − 0.218·21-s + 0.0298·22-s + 0.208·23-s − 0.461·24-s − 0.712·25-s − 0.0792·26-s + 0.192·27-s + 0.303·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.992722997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992722997\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 1.25T + 8T^{2} \) |
| 5 | \( 1 + 5.99T + 125T^{2} \) |
| 11 | \( 1 - 2.45T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.39T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5T + 6.85e3T^{2} \) |
| 29 | \( 1 - 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 529.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 457.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 363.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 216.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 958.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 539.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
| 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
−10.29423041867513086193721348931, −9.685392380120075286063755192442, −8.710088228939149891891499149767, −8.005171703224976994729375740452, −6.95849909659936418745373663648, −5.70279233604739097219231597053, −4.65913639037810878091023170407, −3.70493304574394569934469479517, −2.84523593571486058971162718631, −0.821732305470631423124776655952,
0.821732305470631423124776655952, 2.84523593571486058971162718631, 3.70493304574394569934469479517, 4.65913639037810878091023170407, 5.70279233604739097219231597053, 6.95849909659936418745373663648, 8.005171703224976994729375740452, 8.710088228939149891891499149767, 9.685392380120075286063755192442, 10.29423041867513086193721348931
