| L(s) = 1 | − 1.32·2-s − 0.231·4-s − 5-s − 3.50·7-s + 2.96·8-s + 1.32·10-s + 0.659·11-s + 5.91·13-s + 4.65·14-s − 3.48·16-s − 0.844·17-s + 0.659·19-s + 0.231·20-s − 0.876·22-s + 23-s + 25-s − 7.85·26-s + 0.812·28-s − 1.59·29-s + 6.75·31-s − 1.30·32-s + 1.12·34-s + 3.50·35-s − 11.7·37-s − 0.876·38-s − 2.96·40-s − 6.40·41-s + ⋯ |
| L(s) = 1 | − 0.940·2-s − 0.115·4-s − 0.447·5-s − 1.32·7-s + 1.04·8-s + 0.420·10-s + 0.198·11-s + 1.63·13-s + 1.24·14-s − 0.870·16-s − 0.204·17-s + 0.151·19-s + 0.0518·20-s − 0.186·22-s + 0.208·23-s + 0.200·25-s − 1.54·26-s + 0.153·28-s − 0.295·29-s + 1.21·31-s − 0.230·32-s + 0.192·34-s + 0.592·35-s − 1.93·37-s − 0.142·38-s − 0.469·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 - 0.659T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 + 0.844T + 17T^{2} \) |
| 19 | \( 1 - 0.659T + 19T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 9.47T + 43T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 - 5.78T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
| 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
−9.536459176444697601927410166285, −8.566755606491236533339775438188, −8.290729113865955435769874305499, −6.95209444415982296854284717705, −6.48147426951550840825344547782, −5.19904895479398377103049729230, −3.94986416066646605210272588956, −3.23851115866038196686041551428, −1.41675639146642653919743358536, 0,
1.41675639146642653919743358536, 3.23851115866038196686041551428, 3.94986416066646605210272588956, 5.19904895479398377103049729230, 6.48147426951550840825344547782, 6.95209444415982296854284717705, 8.290729113865955435769874305499, 8.566755606491236533339775438188, 9.536459176444697601927410166285
