| L(s) = 1 | − 2-s − 3.30·3-s + 4-s + 2.30·5-s + 3.30·6-s − 8-s + 7.90·9-s − 2.30·10-s − 2.30·11-s − 3.30·12-s − 1.30·13-s − 7.60·15-s + 16-s + 6·17-s − 7.90·18-s − 2·19-s + 2.30·20-s + 2.30·22-s + 3.90·23-s + 3.30·24-s + 0.302·25-s + 1.30·26-s − 16.2·27-s − 3.90·29-s + 7.60·30-s + 0.302·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.90·3-s + 0.5·4-s + 1.02·5-s + 1.34·6-s − 0.353·8-s + 2.63·9-s − 0.728·10-s − 0.694·11-s − 0.953·12-s − 0.361·13-s − 1.96·15-s + 0.250·16-s + 1.45·17-s − 1.86·18-s − 0.458·19-s + 0.514·20-s + 0.490·22-s + 0.814·23-s + 0.674·24-s + 0.0605·25-s + 0.255·26-s − 3.11·27-s − 0.725·29-s + 1.38·30-s + 0.0543·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 0.302T + 31T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 0.605T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.948020444540585025954223581384, −7.37706904108310094245371423553, −6.47328701984902667661026876569, −6.04490135670116543365577609958, −5.24439581825138048852730723536, −4.87993935682794763504726008572, −3.41380872099491922076615284091, −2.02961284314446325511385898272, −1.18900618583111964745346159764, 0,
1.18900618583111964745346159764, 2.02961284314446325511385898272, 3.41380872099491922076615284091, 4.87993935682794763504726008572, 5.24439581825138048852730723536, 6.04490135670116543365577609958, 6.47328701984902667661026876569, 7.37706904108310094245371423553, 7.948020444540585025954223581384
