| L(s) = 1 | + 2-s − 0.00373·3-s + 4-s − 3.76·5-s − 0.00373·6-s − 2.87·7-s + 8-s − 2.99·9-s − 3.76·10-s − 4.16·11-s − 0.00373·12-s + 2.69·13-s − 2.87·14-s + 0.0140·15-s + 16-s − 2.71·17-s − 2.99·18-s − 3.35·19-s − 3.76·20-s + 0.0107·21-s − 4.16·22-s + 23-s − 0.00373·24-s + 9.14·25-s + 2.69·26-s + 0.0224·27-s − 2.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.00215·3-s + 0.5·4-s − 1.68·5-s − 0.00152·6-s − 1.08·7-s + 0.353·8-s − 0.999·9-s − 1.18·10-s − 1.25·11-s − 0.00107·12-s + 0.748·13-s − 0.768·14-s + 0.00363·15-s + 0.250·16-s − 0.657·17-s − 0.707·18-s − 0.769·19-s − 0.841·20-s + 0.00234·21-s − 0.887·22-s + 0.208·23-s − 0.000763·24-s + 1.82·25-s + 0.529·26-s + 0.00431·27-s − 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4801436716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4801436716\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.00373T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 29 | \( 1 + 8.59T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 4.77T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 8.26T + 83T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.83663249580309884892402922440, −7.46723183039164681201363551820, −6.58133549920021871698376795621, −5.93858327697266288959447173218, −5.15512620811233825522661431448, −4.31780750808494888341555138169, −3.50585094034694272721494221054, −3.19003833685308785454969760049, −2.19650672007463581454982828718, −0.29999087300195304121002930251,
0.29999087300195304121002930251, 2.19650672007463581454982828718, 3.19003833685308785454969760049, 3.50585094034694272721494221054, 4.31780750808494888341555138169, 5.15512620811233825522661431448, 5.93858327697266288959447173218, 6.58133549920021871698376795621, 7.46723183039164681201363551820, 7.83663249580309884892402922440
