L(s) = 1 | − 2-s + 2.00·3-s + 4-s + 2.56·5-s − 2.00·6-s + 7-s − 8-s + 1.03·9-s − 2.56·10-s − 2.73·11-s + 2.00·12-s − 1.28·13-s − 14-s + 5.14·15-s + 16-s − 3.83·17-s − 1.03·18-s − 5.83·19-s + 2.56·20-s + 2.00·21-s + 2.73·22-s − 4.22·23-s − 2.00·24-s + 1.55·25-s + 1.28·26-s − 3.94·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.16·3-s + 0.5·4-s + 1.14·5-s − 0.820·6-s + 0.377·7-s − 0.353·8-s + 0.345·9-s − 0.809·10-s − 0.824·11-s + 0.580·12-s − 0.357·13-s − 0.267·14-s + 1.32·15-s + 0.250·16-s − 0.931·17-s − 0.244·18-s − 1.33·19-s + 0.572·20-s + 0.438·21-s + 0.582·22-s − 0.881·23-s − 0.410·24-s + 0.311·25-s + 0.252·26-s − 0.758·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 2.00T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 0.0674T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 0.938T + 71T^{2} \) |
| 73 | \( 1 + 5.30T + 73T^{2} \) |
| 79 | \( 1 + 5.57T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 + 0.460T + 97T^{2} \) |
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
−7.82409300044693863622691773746, −7.37860052483998077314295292213, −6.22173215402598162435924269146, −5.88082823093622105331740451901, −4.78997192147272741035524533053, −3.92987138315771062566836085053, −2.74671614012987942046776675135, −2.24274013617508248506204718051, −1.73904111707420843247577497946, 0,
1.73904111707420843247577497946, 2.24274013617508248506204718051, 2.74671614012987942046776675135, 3.92987138315771062566836085053, 4.78997192147272741035524533053, 5.88082823093622105331740451901, 6.22173215402598162435924269146, 7.37860052483998077314295292213, 7.82409300044693863622691773746