| L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 11-s − 4·15-s − 4·17-s + 4·19-s + 4·23-s − 25-s − 4·27-s + 2·29-s − 2·31-s + 2·33-s − 6·37-s − 4·41-s + 4·43-s − 2·45-s + 2·47-s − 8·51-s + 2·53-s − 2·55-s + 8·57-s − 6·59-s − 4·61-s + 8·69-s + 12·71-s − 16·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.359·31-s + 0.348·33-s − 0.986·37-s − 0.624·41-s + 0.609·43-s − 0.298·45-s + 0.291·47-s − 1.12·51-s + 0.274·53-s − 0.269·55-s + 1.05·57-s − 0.781·59-s − 0.512·61-s + 0.963·69-s + 1.42·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.35387737260245232732523552443, −7.15309354294377975702479088526, −6.13466945441018037523080365048, −5.22951886432346771455005070243, −4.41035807626250307112773301586, −3.72219842859439930550837717459, −3.14636993455934420027152081733, −2.39127012195943389484667328564, −1.37305950189066983363250464788, 0,
1.37305950189066983363250464788, 2.39127012195943389484667328564, 3.14636993455934420027152081733, 3.72219842859439930550837717459, 4.41035807626250307112773301586, 5.22951886432346771455005070243, 6.13466945441018037523080365048, 7.15309354294377975702479088526, 7.35387737260245232732523552443
