| L(s) = 1 | + 2·5-s + 7-s − 2·13-s + 2·17-s − 4·19-s − 25-s + 2·29-s − 10·31-s + 2·35-s + 8·37-s − 2·41-s − 4·43-s + 6·47-s + 49-s + 12·53-s − 10·59-s − 6·61-s − 4·65-s − 8·67-s − 10·73-s − 4·79-s + 4·83-s + 4·85-s + 2·89-s − 2·91-s − 8·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.371·29-s − 1.79·31-s + 0.338·35-s + 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 1.64·53-s − 1.30·59-s − 0.768·61-s − 0.496·65-s − 0.977·67-s − 1.17·73-s − 0.450·79-s + 0.439·83-s + 0.433·85-s + 0.211·89-s − 0.209·91-s − 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287412673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287412673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.81420411027025, −12.43078403436823, −11.78153303821104, −11.54151111594345, −10.82675909825650, −10.37523248404704, −10.19140853518328, −9.489803087987733, −9.120614201782774, −8.739932198658731, −8.118427682956977, −7.503907931284254, −7.320693883003328, −6.576799579906318, −6.030646141296222, −5.726301940128254, −5.217521630904605, −4.590795553918962, −4.199904719049098, −3.508323079982828, −2.874752204915612, −2.271258336855632, −1.847037858259612, −1.274193844101051, −0.3963807751160910,
0.3963807751160910, 1.274193844101051, 1.847037858259612, 2.271258336855632, 2.874752204915612, 3.508323079982828, 4.199904719049098, 4.590795553918962, 5.217521630904605, 5.726301940128254, 6.030646141296222, 6.576799579906318, 7.320693883003328, 7.503907931284254, 8.118427682956977, 8.739932198658731, 9.120614201782774, 9.489803087987733, 10.19140853518328, 10.37523248404704, 10.82675909825650, 11.54151111594345, 11.78153303821104, 12.43078403436823, 12.81420411027025
