| L(s) = 1 | + 2.33·2-s + 3.43·4-s − 2.03·5-s + 4.28·7-s + 3.34·8-s − 4.74·10-s − 3.57·11-s − 5.73·13-s + 9.98·14-s + 0.920·16-s + 4.13·17-s − 8.31·19-s − 6.99·20-s − 8.33·22-s − 23-s − 0.850·25-s − 13.3·26-s + 14.7·28-s − 29-s + 4.54·31-s − 4.53·32-s + 9.64·34-s − 8.72·35-s − 9.15·37-s − 19.3·38-s − 6.80·40-s − 3.77·41-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 1.71·4-s − 0.910·5-s + 1.61·7-s + 1.18·8-s − 1.50·10-s − 1.07·11-s − 1.59·13-s + 2.66·14-s + 0.230·16-s + 1.00·17-s − 1.90·19-s − 1.56·20-s − 1.77·22-s − 0.208·23-s − 0.170·25-s − 2.62·26-s + 2.77·28-s − 0.185·29-s + 0.816·31-s − 0.801·32-s + 1.65·34-s − 1.47·35-s − 1.50·37-s − 3.14·38-s − 1.07·40-s − 0.589·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + 8.31T + 19T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 5.73T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 0.738T + 97T^{2} \) |
| 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
−7.63153672165505948942175600644, −7.04560771424079117781787378791, −6.02564023452821565772675162227, −5.19438638905264115210947987472, −4.81336903105648660573397392480, −4.30187837237474748128524781758, −3.44801685332102409148316241258, −2.48025195039543559080162730637, −1.87681429596922492256914157204, 0,
1.87681429596922492256914157204, 2.48025195039543559080162730637, 3.44801685332102409148316241258, 4.30187837237474748128524781758, 4.81336903105648660573397392480, 5.19438638905264115210947987472, 6.02564023452821565772675162227, 7.04560771424079117781787378791, 7.63153672165505948942175600644
