| L(s) = 1 | + 5·5-s − 14·7-s − 3·11-s + 47·13-s − 39·17-s − 32·19-s + 99·23-s + 25·25-s + 51·29-s − 83·31-s − 70·35-s + 314·37-s − 108·41-s − 299·43-s − 531·47-s − 147·49-s + 564·53-s − 15·55-s − 12·59-s + 230·61-s + 235·65-s + 268·67-s − 120·71-s + 1.10e3·73-s + 42·77-s + 739·79-s − 1.08e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.0822·11-s + 1.00·13-s − 0.556·17-s − 0.386·19-s + 0.897·23-s + 1/5·25-s + 0.326·29-s − 0.480·31-s − 0.338·35-s + 1.39·37-s − 0.411·41-s − 1.06·43-s − 1.64·47-s − 3/7·49-s + 1.46·53-s − 0.0367·55-s − 0.0264·59-s + 0.482·61-s + 0.448·65-s + 0.488·67-s − 0.200·71-s + 1.77·73-s + 0.0621·77-s + 1.05·79-s − 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.068930994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068930994\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 39 T + p^{3} T^{2} \) |
| 19 | \( 1 + 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 99 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 + 83 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 299 T + p^{3} T^{2} \) |
| 47 | \( 1 + 531 T + p^{3} T^{2} \) |
| 53 | \( 1 - 564 T + p^{3} T^{2} \) |
| 59 | \( 1 + 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1106 T + p^{3} T^{2} \) |
| 79 | \( 1 - 739 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 89 | \( 1 + 120 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1642 T + p^{3} T^{2} \) |
| 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
−8.728763778669797167450134765010, −8.114789582148539371574219374766, −6.89145177749073448588867731458, −6.48263928677127980454363386011, −5.65015448895022212915403007022, −4.73378729975887205175204822435, −3.70966890777271379491093528884, −2.90069963671927878086783572187, −1.82148816433151228547493191468, −0.65075651986814655824313745194,
0.65075651986814655824313745194, 1.82148816433151228547493191468, 2.90069963671927878086783572187, 3.70966890777271379491093528884, 4.73378729975887205175204822435, 5.65015448895022212915403007022, 6.48263928677127980454363386011, 6.89145177749073448588867731458, 8.114789582148539371574219374766, 8.728763778669797167450134765010
