| L(s) = 1 | + 2.39·2-s + 3.75·4-s − 2.64·5-s + 2.83·7-s + 4.21·8-s − 6.34·10-s − 3.73·11-s − 6.68·13-s + 6.81·14-s + 2.59·16-s + 7.52·17-s − 19-s − 9.93·20-s − 8.95·22-s + 3.92·23-s + 1.99·25-s − 16.0·26-s + 10.6·28-s + 3.46·29-s − 7.34·31-s − 2.19·32-s + 18.0·34-s − 7.51·35-s − 0.801·37-s − 2.39·38-s − 11.1·40-s − 3.88·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.87·4-s − 1.18·5-s + 1.07·7-s + 1.48·8-s − 2.00·10-s − 1.12·11-s − 1.85·13-s + 1.82·14-s + 0.648·16-s + 1.82·17-s − 0.229·19-s − 2.22·20-s − 1.90·22-s + 0.819·23-s + 0.399·25-s − 3.14·26-s + 2.01·28-s + 0.643·29-s − 1.31·31-s − 0.388·32-s + 3.09·34-s − 1.26·35-s − 0.131·37-s − 0.389·38-s − 1.76·40-s − 0.606·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 - 2.83T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 6.68T + 13T^{2} \) |
| 17 | \( 1 - 7.52T + 17T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 0.801T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 59 | \( 1 + 0.0753T + 59T^{2} \) |
| 61 | \( 1 - 7.67T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 9.00T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 9.93T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
| 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
−7.39541470449787429219916960835, −6.90767346109087538984621054873, −5.61755693228428854194698063223, −5.17427875068367201225445752919, −4.79049036103239305233526313015, −4.05226571888969241253617603055, −3.19644378792383118288882181959, −2.68590365125824488073292512095, −1.63340942963082875270976772078, 0,
1.63340942963082875270976772078, 2.68590365125824488073292512095, 3.19644378792383118288882181959, 4.05226571888969241253617603055, 4.79049036103239305233526313015, 5.17427875068367201225445752919, 5.61755693228428854194698063223, 6.90767346109087538984621054873, 7.39541470449787429219916960835
