| L(s) = 1 | + 2-s + 4-s + 3.31·5-s − 3.41·7-s + 8-s + 3.31·10-s + 3.88·11-s − 0.659·13-s − 3.41·14-s + 16-s + 1.70·17-s + 2.86·19-s + 3.31·20-s + 3.88·22-s + 3.18·23-s + 6.01·25-s − 0.659·26-s − 3.41·28-s + 1.15·29-s − 0.368·31-s + 32-s + 1.70·34-s − 11.3·35-s + 5.75·37-s + 2.86·38-s + 3.31·40-s − 6.68·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.48·5-s − 1.29·7-s + 0.353·8-s + 1.04·10-s + 1.17·11-s − 0.183·13-s − 0.913·14-s + 0.250·16-s + 0.413·17-s + 0.657·19-s + 0.742·20-s + 0.827·22-s + 0.663·23-s + 1.20·25-s − 0.129·26-s − 0.646·28-s + 0.214·29-s − 0.0661·31-s + 0.176·32-s + 0.292·34-s − 1.91·35-s + 0.946·37-s + 0.465·38-s + 0.524·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.493381080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.493381080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 + 0.659T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + 0.368T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 + 0.592T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 - 1.63T + 71T^{2} \) |
| 73 | \( 1 + 0.306T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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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
−7.55917025642741073208283545352, −6.76475963596601002684283908472, −6.39889162328407829370654766378, −5.81392882069575758222582916215, −5.19496246496872996675990618469, −4.29085773631794015101424052903, −3.36132640367920447439099166799, −2.85797222305635053967439103192, −1.89200885746742275281247954051, −0.985815341242616477174204183899,
0.985815341242616477174204183899, 1.89200885746742275281247954051, 2.85797222305635053967439103192, 3.36132640367920447439099166799, 4.29085773631794015101424052903, 5.19496246496872996675990618469, 5.81392882069575758222582916215, 6.39889162328407829370654766378, 6.76475963596601002684283908472, 7.55917025642741073208283545352
