| L(s) = 1 | + 1.56·2-s + 3-s + 0.438·4-s + 3.56·5-s + 1.56·6-s − 0.561·7-s − 2.43·8-s + 9-s + 5.56·10-s + 2·11-s + 0.438·12-s − 0.876·14-s + 3.56·15-s − 4.68·16-s − 1.56·17-s + 1.56·18-s − 7.12·19-s + 1.56·20-s − 0.561·21-s + 3.12·22-s + 2·23-s − 2.43·24-s + 7.68·25-s + 27-s − 0.246·28-s + 6.68·29-s + 5.56·30-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.219·4-s + 1.59·5-s + 0.637·6-s − 0.212·7-s − 0.862·8-s + 0.333·9-s + 1.75·10-s + 0.603·11-s + 0.126·12-s − 0.234·14-s + 0.919·15-s − 1.17·16-s − 0.378·17-s + 0.368·18-s − 1.63·19-s + 0.349·20-s − 0.122·21-s + 0.665·22-s + 0.417·23-s − 0.497·24-s + 1.53·25-s + 0.192·27-s − 0.0465·28-s + 1.24·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.230909492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.230909492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−10.86746015591716693988632929975, −9.937095145303307375204410934467, −9.131413003359716976605778254528, −8.531346012746000623099240922277, −6.71600125421775928105928531431, −6.25886535335637483940327810364, −5.18038228423165673520942204067, −4.22577094883249025782498773303, −2.99142665580051132207938482349, −1.93481065374349873308465218165,
1.93481065374349873308465218165, 2.99142665580051132207938482349, 4.22577094883249025782498773303, 5.18038228423165673520942204067, 6.25886535335637483940327810364, 6.71600125421775928105928531431, 8.531346012746000623099240922277, 9.131413003359716976605778254528, 9.937095145303307375204410934467, 10.86746015591716693988632929975
