| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s − 6·11-s + 13-s − 14-s + 16-s + 3·17-s + 2·19-s + 3·20-s − 6·22-s + 4·25-s + 26-s − 28-s − 6·29-s − 4·31-s + 32-s + 3·34-s − 3·35-s − 7·37-s + 2·38-s + 3·40-s − 43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 1.80·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.670·20-s − 1.27·22-s + 4/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1.15·37-s + 0.324·38-s + 0.474·40-s − 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008799896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008799896\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.57399662018781984180868180363, −11.16559240112924374874969825279, −10.25096846899164676242515061975, −9.569105792671075621984918394603, −8.109212031498814193717580943346, −6.94019412385488865960361148937, −5.66417880764699758598698858020, −5.24164848449732929783448594193, −3.34467457020301574582723277384, −2.09859100780083292577684604843,
2.09859100780083292577684604843, 3.34467457020301574582723277384, 5.24164848449732929783448594193, 5.66417880764699758598698858020, 6.94019412385488865960361148937, 8.109212031498814193717580943346, 9.569105792671075621984918394603, 10.25096846899164676242515061975, 11.16559240112924374874969825279, 12.57399662018781984180868180363
