| L(s) = 1 | + 1.81·2-s + 0.934·3-s + 1.29·4-s − 2.29·5-s + 1.69·6-s − 1.28·8-s − 2.12·9-s − 4.16·10-s − 4.58·11-s + 1.20·12-s + 4.23·13-s − 2.14·15-s − 4.91·16-s + 2.72·17-s − 3.85·18-s − 2.91·19-s − 2.96·20-s − 8.32·22-s − 3.56·23-s − 1.19·24-s + 0.278·25-s + 7.69·26-s − 4.79·27-s − 5.77·29-s − 3.89·30-s + 31-s − 6.34·32-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.539·3-s + 0.646·4-s − 1.02·5-s + 0.692·6-s − 0.454·8-s − 0.709·9-s − 1.31·10-s − 1.38·11-s + 0.348·12-s + 1.17·13-s − 0.554·15-s − 1.22·16-s + 0.659·17-s − 0.909·18-s − 0.667·19-s − 0.663·20-s − 1.77·22-s − 0.743·23-s − 0.244·24-s + 0.0556·25-s + 1.50·26-s − 0.921·27-s − 1.07·29-s − 0.711·30-s + 0.179·31-s − 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 0.934T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 6.69T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 9.13T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−8.815598067747221871556584213409, −8.131419511815179657878404435480, −7.64436151231233474003787626084, −6.30076095510541277108673745749, −5.66337129209814203360478177429, −4.76819171775906041593397474894, −3.76072455699305343912432007512, −3.32506916869790037473351175272, −2.29106773198877467612437230483, 0,
2.29106773198877467612437230483, 3.32506916869790037473351175272, 3.76072455699305343912432007512, 4.76819171775906041593397474894, 5.66337129209814203360478177429, 6.30076095510541277108673745749, 7.64436151231233474003787626084, 8.131419511815179657878404435480, 8.815598067747221871556584213409
