| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 12-s − 13-s − 14-s + 16-s − 17-s + 19-s + 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + 32-s − 34-s + 2·37-s + 38-s + 39-s + 42-s − 46-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 12-s − 13-s − 14-s + 16-s − 17-s + 19-s + 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + 32-s − 34-s + 2·37-s + 38-s + 39-s + 42-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7788472013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7788472013\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| 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
−13.03984448983643196413376115173, −12.29223974961841780502300444212, −11.48149381049958660012230602000, −10.56096522015558538979777339827, −9.420874252673057307646955909800, −7.50781507085966449620544772843, −6.46055524279898288954266410418, −5.63050345906219303013958354741, −4.44276652764072509073740472545, −2.81196093969486835641282132236,
2.81196093969486835641282132236, 4.44276652764072509073740472545, 5.63050345906219303013958354741, 6.46055524279898288954266410418, 7.50781507085966449620544772843, 9.420874252673057307646955909800, 10.56096522015558538979777339827, 11.48149381049958660012230602000, 12.29223974961841780502300444212, 13.03984448983643196413376115173
