| L(s) = 1 | + 5-s + 2·7-s + 2·11-s − 2·13-s − 4·17-s + 25-s + 2·29-s + 8·31-s + 2·35-s + 8·37-s + 8·41-s + 43-s + 8·47-s − 3·49-s − 14·53-s + 2·55-s + 10·59-s − 12·61-s − 2·65-s + 12·67-s + 8·71-s − 2·73-s + 4·77-s − 4·79-s + 6·83-s − 4·85-s + 18·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s − 0.970·17-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.338·35-s + 1.31·37-s + 1.24·41-s + 0.152·43-s + 1.16·47-s − 3/7·49-s − 1.92·53-s + 0.269·55-s + 1.30·59-s − 1.53·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s + 0.658·83-s − 0.433·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.019255229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.019255229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.94976758007421, −14.50450737401240, −14.17663454565791, −13.51002930036935, −13.09703667113490, −12.32692823403070, −12.00739266501462, −11.19041105660351, −11.01537175012936, −10.27373352785673, −9.576932077824187, −9.274615524929738, −8.596753515409412, −7.938741379296974, −7.575360244569740, −6.532916444294402, −6.462323837053261, −5.605826468184498, −4.832172359678471, −4.493288352824493, −3.808973203878179, −2.732650069778797, −2.334281615829102, −1.455271808609235, −0.6925083591565823,
0.6925083591565823, 1.455271808609235, 2.334281615829102, 2.732650069778797, 3.808973203878179, 4.493288352824493, 4.832172359678471, 5.605826468184498, 6.462323837053261, 6.532916444294402, 7.575360244569740, 7.938741379296974, 8.596753515409412, 9.274615524929738, 9.576932077824187, 10.27373352785673, 11.01537175012936, 11.19041105660351, 12.00739266501462, 12.32692823403070, 13.09703667113490, 13.51002930036935, 14.17663454565791, 14.50450737401240, 14.94976758007421
