| L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s − 3·9-s + 4·10-s + 11-s + 14-s + 16-s − 4·17-s − 3·18-s + 6·19-s + 4·20-s + 22-s + 4·23-s + 11·25-s + 28-s − 2·29-s + 2·31-s + 32-s − 4·34-s + 4·35-s − 3·36-s − 10·37-s + 6·38-s + 4·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s − 9-s + 1.26·10-s + 0.301·11-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 1.37·19-s + 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s + 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s − 0.685·34-s + 0.676·35-s − 1/2·36-s − 1.64·37-s + 0.973·38-s + 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.894831827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.894831827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.16552178060692, −14.57031762039639, −14.15461261102550, −13.67670012107861, −13.42523278521620, −12.84898662806723, −12.05715833544776, −11.59313254955678, −11.07413486354401, −10.44815255040377, −9.937663274798511, −9.273697500080707, −8.807217398023782, −8.290437864524924, −7.238210248617225, −6.789133122701942, −6.187166406388450, −5.635094538681812, −5.053372426456164, −4.840906697171069, −3.531208793219260, −3.095466398386260, −2.206594576147613, −1.828880868566011, −0.8511179377121311,
0.8511179377121311, 1.828880868566011, 2.206594576147613, 3.095466398386260, 3.531208793219260, 4.840906697171069, 5.053372426456164, 5.635094538681812, 6.187166406388450, 6.789133122701942, 7.238210248617225, 8.290437864524924, 8.807217398023782, 9.273697500080707, 9.937663274798511, 10.44815255040377, 11.07413486354401, 11.59313254955678, 12.05715833544776, 12.84898662806723, 13.42523278521620, 13.67670012107861, 14.15461261102550, 14.57031762039639, 15.16552178060692
