| L(s) = 1 | + 0.685·3-s − 1.94·5-s − 1.34·7-s − 2.53·9-s + 3.60·11-s + 3.72·13-s − 1.33·15-s − 6.00·17-s + 5.57·19-s − 0.925·21-s − 5.27·23-s − 1.21·25-s − 3.79·27-s + 5.26·29-s + 7.76·31-s + 2.46·33-s + 2.62·35-s − 5.58·37-s + 2.55·39-s − 6.52·41-s + 6.65·43-s + 4.92·45-s + 1.94·47-s − 5.17·49-s − 4.11·51-s + 9.28·53-s − 7.00·55-s + ⋯ |
| L(s) = 1 | + 0.395·3-s − 0.870·5-s − 0.510·7-s − 0.843·9-s + 1.08·11-s + 1.03·13-s − 0.344·15-s − 1.45·17-s + 1.27·19-s − 0.201·21-s − 1.10·23-s − 0.242·25-s − 0.729·27-s + 0.977·29-s + 1.39·31-s + 0.429·33-s + 0.444·35-s − 0.917·37-s + 0.408·39-s − 1.01·41-s + 1.01·43-s + 0.734·45-s + 0.283·47-s − 0.739·49-s − 0.576·51-s + 1.27·53-s − 0.945·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 0.685T + 3T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 9.28T + 53T^{2} \) |
| 59 | \( 1 + 3.36T + 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 6.36T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
| 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
−7.61951587194577187573535846069, −6.61852545493261486434147400313, −6.35845405436361458277201030967, −5.45406894274941615589680485610, −4.40690908858306013198677984876, −3.81572156719664186189910640736, −3.25372897814499045847818280663, −2.37558842212169810243770991956, −1.18454682121919665563918139485, 0,
1.18454682121919665563918139485, 2.37558842212169810243770991956, 3.25372897814499045847818280663, 3.81572156719664186189910640736, 4.40690908858306013198677984876, 5.45406894274941615589680485610, 6.35845405436361458277201030967, 6.61852545493261486434147400313, 7.61951587194577187573535846069
