| L(s) = 1 | + 5-s − 4·7-s − 3·9-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s + 6·37-s − 6·41-s − 8·43-s − 3·45-s + 4·47-s + 9·49-s + 6·53-s + 4·55-s − 4·59-s − 2·61-s + 12·63-s − 2·65-s + 8·67-s − 6·73-s − 16·77-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s − 0.702·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7422062367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7422062367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−16.54367915654916380177508640952, −14.95092089654112303026933531279, −13.90086992520560888964063455546, −12.70951559228550877361223816630, −11.52426214006865205005559601963, −9.851840183892789826086741161768, −9.022035638415746178025932660186, −6.97238544391138654785050907556, −5.70093888866808737514223874840, −3.26180587408034592610487247010,
3.26180587408034592610487247010, 5.70093888866808737514223874840, 6.97238544391138654785050907556, 9.022035638415746178025932660186, 9.851840183892789826086741161768, 11.52426214006865205005559601963, 12.70951559228550877361223816630, 13.90086992520560888964063455546, 14.95092089654112303026933531279, 16.54367915654916380177508640952
