| L(s) = 1 | + 1.86·2-s + 14.8·3-s − 28.5·4-s + 27.6·6-s − 202.·7-s − 112.·8-s − 21.8·9-s + 436.·11-s − 424.·12-s + 617.·13-s − 377.·14-s + 703.·16-s − 833.·17-s − 40.6·18-s + 1.35e3·19-s − 3.01e3·21-s + 812.·22-s + 904.·23-s − 1.67e3·24-s + 1.14e3·26-s − 3.93e3·27-s + 5.79e3·28-s + 5.32e3·29-s + 919.·31-s + 4.91e3·32-s + 6.49e3·33-s − 1.55e3·34-s + ⋯ |
| L(s) = 1 | + 0.328·2-s + 0.954·3-s − 0.891·4-s + 0.313·6-s − 1.56·7-s − 0.622·8-s − 0.0898·9-s + 1.08·11-s − 0.850·12-s + 1.01·13-s − 0.514·14-s + 0.687·16-s − 0.699·17-s − 0.0295·18-s + 0.862·19-s − 1.49·21-s + 0.357·22-s + 0.356·23-s − 0.593·24-s + 0.333·26-s − 1.03·27-s + 1.39·28-s + 1.17·29-s + 0.171·31-s + 0.848·32-s + 1.03·33-s − 0.230·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 1.86T + 32T^{2} \) |
| 3 | \( 1 - 14.8T + 243T^{2} \) |
| 7 | \( 1 + 202.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 436.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 617.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 833.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 904.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 919.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.93e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.79e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.675680358202523813164104068019, −8.372960746218058435091505816841, −6.86304270771407461321059478553, −6.30566952947924391133668857801, −5.26848178339918152243624928139, −3.96833511422529370874116132483, −3.49174115856135652421735193474, −2.76224240947731683711911674034, −1.16227305974209231460088129577, 0,
1.16227305974209231460088129577, 2.76224240947731683711911674034, 3.49174115856135652421735193474, 3.96833511422529370874116132483, 5.26848178339918152243624928139, 6.30566952947924391133668857801, 6.86304270771407461321059478553, 8.372960746218058435091505816841, 8.675680358202523813164104068019
