| L(s) = 1 | + 4·4-s + 5·5-s + 7·7-s + 13·11-s − 19·13-s + 16·16-s − 29·17-s + 20·20-s + 25·25-s + 28·28-s − 23·29-s + 35·35-s + 52·44-s + 31·47-s + 49·49-s − 76·52-s + 65·55-s + 64·64-s − 95·65-s − 116·68-s − 2·71-s − 34·73-s + 91·77-s − 157·79-s + 80·80-s − 86·83-s − 145·85-s + ⋯ |
| L(s) = 1 | + 4-s + 5-s + 7-s + 1.18·11-s − 1.46·13-s + 16-s − 1.70·17-s + 20-s + 25-s + 28-s − 0.793·29-s + 35-s + 1.18·44-s + 0.659·47-s + 49-s − 1.46·52-s + 1.18·55-s + 64-s − 1.46·65-s − 1.70·68-s − 0.0281·71-s − 0.465·73-s + 1.18·77-s − 1.98·79-s + 80-s − 1.03·83-s − 1.70·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.590335103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.590335103\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 13 T + p^{2} T^{2} \) |
| 13 | \( 1 + 19 T + p^{2} T^{2} \) |
| 17 | \( 1 + 29 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 23 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 31 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( 1 + 2 T + p^{2} T^{2} \) |
| 73 | \( 1 + 34 T + p^{2} T^{2} \) |
| 79 | \( 1 + 157 T + p^{2} T^{2} \) |
| 83 | \( 1 + 86 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 149 T + p^{2} T^{2} \) |
| 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
−11.41471185348705879677928669267, −10.64084617526008073011918226905, −9.605669241673241644487024695075, −8.708144358528762463155053103260, −7.35424748215308995712873134706, −6.65245416775561525773704554937, −5.56953187060849358647447266138, −4.40216654259092460050681315403, −2.50750673818329292653414605181, −1.65077545698102342480019274080,
1.65077545698102342480019274080, 2.50750673818329292653414605181, 4.40216654259092460050681315403, 5.56953187060849358647447266138, 6.65245416775561525773704554937, 7.35424748215308995712873134706, 8.708144358528762463155053103260, 9.605669241673241644487024695075, 10.64084617526008073011918226905, 11.41471185348705879677928669267
