| L(s) = 1 | + 0.236·2-s − 2.87·3-s − 1.94·4-s − 5-s − 0.677·6-s + 1.02·7-s − 0.931·8-s + 5.24·9-s − 0.236·10-s + 0.916·11-s + 5.58·12-s + 4.99·13-s + 0.241·14-s + 2.87·15-s + 3.66·16-s + 0.345·17-s + 1.23·18-s − 5.30·19-s + 1.94·20-s − 2.93·21-s + 0.216·22-s − 7.21·23-s + 2.67·24-s + 25-s + 1.17·26-s − 6.44·27-s − 1.98·28-s + ⋯ |
| L(s) = 1 | + 0.166·2-s − 1.65·3-s − 0.972·4-s − 0.447·5-s − 0.276·6-s + 0.386·7-s − 0.329·8-s + 1.74·9-s − 0.0746·10-s + 0.276·11-s + 1.61·12-s + 1.38·13-s + 0.0644·14-s + 0.741·15-s + 0.917·16-s + 0.0838·17-s + 0.291·18-s − 1.21·19-s + 0.434·20-s − 0.640·21-s + 0.0461·22-s − 1.50·23-s + 0.545·24-s + 0.200·25-s + 0.231·26-s − 1.23·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 0.236T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 - 0.916T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 0.345T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 7.21T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 - 9.99T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 9.72T + 67T^{2} \) |
| 71 | \( 1 - 0.435T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 + 4.36T + 79T^{2} \) |
| 83 | \( 1 + 9.26T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
| 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
−9.971418702623043891700848736214, −8.452433518686175009157825461961, −8.204767947991794871108089560919, −6.50687939541780888295466074856, −6.23674441840629068922137331081, −5.08210146811284364005604198164, −4.47715637092761514735997251787, −3.63547080183708698198608749544, −1.33538730130272320767838242290, 0,
1.33538730130272320767838242290, 3.63547080183708698198608749544, 4.47715637092761514735997251787, 5.08210146811284364005604198164, 6.23674441840629068922137331081, 6.50687939541780888295466074856, 8.204767947991794871108089560919, 8.452433518686175009157825461961, 9.971418702623043891700848736214
