| L(s) = 1 | + 2-s − 2.52·3-s + 4-s + 2.27·5-s − 2.52·6-s + 1.33·7-s + 8-s + 3.38·9-s + 2.27·10-s + 2.72·11-s − 2.52·12-s + 1.63·13-s + 1.33·14-s − 5.74·15-s + 16-s − 5.34·17-s + 3.38·18-s + 4.68·19-s + 2.27·20-s − 3.36·21-s + 2.72·22-s − 23-s − 2.52·24-s + 0.174·25-s + 1.63·26-s − 0.968·27-s + 1.33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.45·3-s + 0.5·4-s + 1.01·5-s − 1.03·6-s + 0.503·7-s + 0.353·8-s + 1.12·9-s + 0.719·10-s + 0.820·11-s − 0.729·12-s + 0.454·13-s + 0.355·14-s − 1.48·15-s + 0.250·16-s − 1.29·17-s + 0.797·18-s + 1.07·19-s + 0.508·20-s − 0.733·21-s + 0.580·22-s − 0.208·23-s − 0.515·24-s + 0.0349·25-s + 0.321·26-s − 0.186·27-s + 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.157655497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157655497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 + 6.72T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 6.26T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−9.801162309414136742003951576622, −9.014038704118341229366630103845, −7.74628663849771514650666911117, −6.56220689811321189206201330807, −6.30542338923103459121458949871, −5.41998281708017933650216930483, −4.81968628466994079737142232406, −3.82751259500434054604669552351, −2.24445137238024824657481029161, −1.12947347764198509471219618012,
1.12947347764198509471219618012, 2.24445137238024824657481029161, 3.82751259500434054604669552351, 4.81968628466994079737142232406, 5.41998281708017933650216930483, 6.30542338923103459121458949871, 6.56220689811321189206201330807, 7.74628663849771514650666911117, 9.014038704118341229366630103845, 9.801162309414136742003951576622
