| L(s) = 1 | + 5-s + 7-s − 3·9-s + 3·11-s + 4·13-s − 3·17-s + 19-s + 8·23-s − 4·25-s − 2·31-s + 35-s + 8·37-s + 11·43-s − 3·45-s + 7·47-s − 6·49-s − 2·53-s + 3·55-s + 6·59-s + 61-s − 3·63-s + 4·65-s − 10·67-s − 2·71-s + 5·73-s + 3·77-s + 2·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s + 0.904·11-s + 1.10·13-s − 0.727·17-s + 0.229·19-s + 1.66·23-s − 4/5·25-s − 0.359·31-s + 0.169·35-s + 1.31·37-s + 1.67·43-s − 0.447·45-s + 1.02·47-s − 6/7·49-s − 0.274·53-s + 0.404·55-s + 0.781·59-s + 0.128·61-s − 0.377·63-s + 0.496·65-s − 1.22·67-s − 0.237·71-s + 0.585·73-s + 0.341·77-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911044309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911044309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.444375464347950216129902591841, −9.013850528711372623404088298718, −8.265849533784556159385578271594, −7.22663142743355843577020555382, −6.23231971111290599279730913097, −5.69887754666040376701153368082, −4.57069618429307251486414062571, −3.55029150041215641334184982487, −2.41972145972916601985658200179, −1.10874592215816899307332244272,
1.10874592215816899307332244272, 2.41972145972916601985658200179, 3.55029150041215641334184982487, 4.57069618429307251486414062571, 5.69887754666040376701153368082, 6.23231971111290599279730913097, 7.22663142743355843577020555382, 8.265849533784556159385578271594, 9.013850528711372623404088298718, 9.444375464347950216129902591841
