| L(s) = 1 | − 2.42·2-s − 2.96·3-s + 3.90·4-s − 5-s + 7.19·6-s + 2.52·7-s − 4.61·8-s + 5.77·9-s + 2.42·10-s + 11-s − 11.5·12-s + 4.42·13-s − 6.13·14-s + 2.96·15-s + 3.41·16-s − 1.29·17-s − 14.0·18-s + 4.83·19-s − 3.90·20-s − 7.48·21-s − 2.42·22-s − 0.787·23-s + 13.6·24-s + 25-s − 10.7·26-s − 8.23·27-s + 9.85·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.71·3-s + 1.95·4-s − 0.447·5-s + 2.93·6-s + 0.955·7-s − 1.63·8-s + 1.92·9-s + 0.768·10-s + 0.301·11-s − 3.33·12-s + 1.22·13-s − 1.64·14-s + 0.765·15-s + 0.852·16-s − 0.315·17-s − 3.30·18-s + 1.10·19-s − 0.872·20-s − 1.63·21-s − 0.517·22-s − 0.164·23-s + 2.79·24-s + 0.200·25-s − 2.10·26-s − 1.58·27-s + 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 + 0.787T + 23T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 3.51T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 + 3.99T + 61T^{2} \) |
| 67 | \( 1 - 6.32T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 0.538T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
| 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
−8.309391037338290482219317135516, −7.37782878618781546626367552440, −6.67357853628618378122190764376, −6.24912718157245965091736964213, −5.16155774088622774160699119275, −4.58222060158875127713219466135, −3.29211766857613945402517189980, −1.58907456002149899532273147170, −1.15990927723402156402446529862, 0,
1.15990927723402156402446529862, 1.58907456002149899532273147170, 3.29211766857613945402517189980, 4.58222060158875127713219466135, 5.16155774088622774160699119275, 6.24912718157245965091736964213, 6.67357853628618378122190764376, 7.37782878618781546626367552440, 8.309391037338290482219317135516
