| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 9-s − 2·11-s − 2·13-s + 4·15-s + 4·19-s − 4·21-s − 2·23-s − 25-s − 4·27-s + 2·29-s + 10·31-s − 4·33-s − 4·35-s + 10·37-s − 4·39-s − 2·41-s + 4·43-s + 2·45-s − 3·49-s − 6·53-s − 4·55-s + 8·57-s + 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.917·19-s − 0.872·21-s − 0.417·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s + 1.79·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.342406638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.342406638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.69385401813515, −15.24077177517039, −14.47828426554317, −14.09189960385583, −13.65817636447881, −13.13965240930328, −12.76005163047924, −11.89830281027485, −11.45483731808908, −10.41797178349914, −10.00116381359280, −9.565238175294130, −9.175723347224774, −8.392793631667120, −7.804691852460717, −7.426294905446299, −6.343330346765449, −6.126554271401464, −5.237009129677293, −4.580629354882899, −3.669111283461761, −2.921440608573427, −2.585332619187979, −1.854821754904921, −0.6937315394327215,
0.6937315394327215, 1.854821754904921, 2.585332619187979, 2.921440608573427, 3.669111283461761, 4.580629354882899, 5.237009129677293, 6.126554271401464, 6.343330346765449, 7.426294905446299, 7.804691852460717, 8.392793631667120, 9.175723347224774, 9.565238175294130, 10.00116381359280, 10.41797178349914, 11.45483731808908, 11.89830281027485, 12.76005163047924, 13.13965240930328, 13.65817636447881, 14.09189960385583, 14.47828426554317, 15.24077177517039, 15.69385401813515
