| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3·7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s + 3·14-s + 15-s + 16-s − 17-s + 18-s − 4·19-s + 20-s + 3·21-s + 4·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.654·21-s + 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.767774340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.767774340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.57312661566045, −12.02221005855584, −11.64550177690840, −11.32025194016557, −10.61478571167890, −10.31426465145917, −9.941684483890672, −9.081262129758052, −8.836088691508911, −8.485421865330943, −7.898689240107004, −7.425995745620367, −6.922526050573438, −6.325437813931816, −6.092130186320955, −5.469059242406120, −4.871356616014648, −4.312086524434192, −4.147830411495673, −3.529907055324695, −2.885316172231114, −2.220572250791029, −1.847109100420537, −1.400671308362440, −0.6567373115301360,
0.6567373115301360, 1.400671308362440, 1.847109100420537, 2.220572250791029, 2.885316172231114, 3.529907055324695, 4.147830411495673, 4.312086524434192, 4.871356616014648, 5.469059242406120, 6.092130186320955, 6.325437813931816, 6.922526050573438, 7.425995745620367, 7.898689240107004, 8.485421865330943, 8.836088691508911, 9.081262129758052, 9.941684483890672, 10.31426465145917, 10.61478571167890, 11.32025194016557, 11.64550177690840, 12.02221005855584, 12.57312661566045
