| L(s) = 1 | + 1.06·2-s + 3-s − 0.862·4-s − 5-s + 1.06·6-s + 1.23·7-s − 3.05·8-s + 9-s − 1.06·10-s − 6.35·11-s − 0.862·12-s − 0.420·13-s + 1.31·14-s − 15-s − 1.52·16-s + 6.90·17-s + 1.06·18-s − 7.45·19-s + 0.862·20-s + 1.23·21-s − 6.77·22-s − 7.93·23-s − 3.05·24-s + 25-s − 0.448·26-s + 27-s − 1.06·28-s + ⋯ |
| L(s) = 1 | + 0.754·2-s + 0.577·3-s − 0.431·4-s − 0.447·5-s + 0.435·6-s + 0.465·7-s − 1.07·8-s + 0.333·9-s − 0.337·10-s − 1.91·11-s − 0.249·12-s − 0.116·13-s + 0.350·14-s − 0.258·15-s − 0.382·16-s + 1.67·17-s + 0.251·18-s − 1.71·19-s + 0.192·20-s + 0.268·21-s − 1.44·22-s − 1.65·23-s − 0.623·24-s + 0.200·25-s − 0.0880·26-s + 0.192·27-s − 0.200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.089928215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089928215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 + 0.420T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 8.14T + 31T^{2} \) |
| 37 | \( 1 - 0.961T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 5.30T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 + 1.91T + 97T^{2} \) |
| 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
−8.130210990330861409431692739990, −7.69589941658634087903250568947, −6.53996402898239507499053866995, −5.66809583369718447496960978091, −5.17000384815350599578402957442, −4.29508185272625708962191095293, −3.86457998943965442283255622682, −2.79731982215794583876721254501, −2.30229083800241424941375650924, −0.63520514960573146582714302008,
0.63520514960573146582714302008, 2.30229083800241424941375650924, 2.79731982215794583876721254501, 3.86457998943965442283255622682, 4.29508185272625708962191095293, 5.17000384815350599578402957442, 5.66809583369718447496960978091, 6.53996402898239507499053866995, 7.69589941658634087903250568947, 8.130210990330861409431692739990
