| L(s) = 1 | + 2.51·2-s − 1.07·3-s + 4.30·4-s − 5-s − 2.71·6-s − 7-s + 5.77·8-s − 1.83·9-s − 2.51·10-s + 5.33·11-s − 4.64·12-s − 4.76·13-s − 2.51·14-s + 1.07·15-s + 5.90·16-s + 0.368·17-s − 4.60·18-s + 2.49·19-s − 4.30·20-s + 1.07·21-s + 13.3·22-s − 5.15·23-s − 6.23·24-s + 25-s − 11.9·26-s + 5.22·27-s − 4.30·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.623·3-s + 2.15·4-s − 0.447·5-s − 1.10·6-s − 0.377·7-s + 2.04·8-s − 0.611·9-s − 0.793·10-s + 1.60·11-s − 1.34·12-s − 1.32·13-s − 0.670·14-s + 0.278·15-s + 1.47·16-s + 0.0893·17-s − 1.08·18-s + 0.572·19-s − 0.961·20-s + 0.235·21-s + 2.85·22-s − 1.07·23-s − 1.27·24-s + 0.200·25-s − 2.34·26-s + 1.00·27-s − 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 + 1.07T + 3T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 - 0.368T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 6.93T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 6.01T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 - 8.00T + 71T^{2} \) |
| 73 | \( 1 + 9.97T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 1.33T + 83T^{2} \) |
| 89 | \( 1 - 6.69T + 89T^{2} \) |
| 97 | \( 1 + 7.62T + 97T^{2} \) |
| 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
−7.07401920133662066705869060741, −6.54968662688486456677039841480, −6.00677868644892091291636048567, −5.28994563283613662466590475639, −4.72296061017621632343098582217, −3.95988529393430414490185078547, −3.38363513751567393117751213199, −2.60481360417179968465997071338, −1.55601153382548051136461385111, 0,
1.55601153382548051136461385111, 2.60481360417179968465997071338, 3.38363513751567393117751213199, 3.95988529393430414490185078547, 4.72296061017621632343098582217, 5.28994563283613662466590475639, 6.00677868644892091291636048567, 6.54968662688486456677039841480, 7.07401920133662066705869060741
