| L(s) = 1 | + 2·2-s + 2·4-s − 12·5-s + 10·7-s + 17·9-s − 24·10-s + 24·13-s + 20·14-s − 4·16-s − 26·17-s + 34·18-s + 2·19-s − 24·20-s + 34·23-s + 72·25-s + 48·26-s + 20·28-s − 38·29-s + 72·31-s − 8·32-s − 52·34-s − 120·35-s + 34·36-s + 4·38-s − 96·43-s − 204·45-s + 68·46-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 2.39·5-s + 10/7·7-s + 17/9·9-s − 2.39·10-s + 1.84·13-s + 10/7·14-s − 1/4·16-s − 1.52·17-s + 17/9·18-s + 2/19·19-s − 6/5·20-s + 1.47·23-s + 2.87·25-s + 1.84·26-s + 5/7·28-s − 1.31·29-s + 2.32·31-s − 1/4·32-s − 1.52·34-s − 3.42·35-s + 0.944·36-s + 2/19·38-s − 2.23·43-s − 4.53·45-s + 1.47·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.125532245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125532245\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 217 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T + 338 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2737 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 55 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 108 T + 5832 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1922 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1223 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T + 3362 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 156 T + 12168 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.01197277949950273024189208868, −13.73658811430443438422004896969, −13.67849113108650617955798322613, −12.89953516361972655082105256017, −12.49280519255927801192241926355, −11.82621272311692655022977411977, −11.26174674038313996011824309538, −11.12944737388894344324574609371, −10.62002553710164858132584906439, −9.447031656151791161305840763762, −8.490585781888535812080098651019, −8.268430200691675953502512478872, −7.53843871147502139961644112646, −7.00076492493366872559263307141, −6.36434110504305487495097692804, −4.89732584906448357170786767790, −4.50182781854935573580767316892, −4.08449534222092914015461477607, −3.35282969934221222499984325349, −1.38278432714208765773967501595,
1.38278432714208765773967501595, 3.35282969934221222499984325349, 4.08449534222092914015461477607, 4.50182781854935573580767316892, 4.89732584906448357170786767790, 6.36434110504305487495097692804, 7.00076492493366872559263307141, 7.53843871147502139961644112646, 8.268430200691675953502512478872, 8.490585781888535812080098651019, 9.447031656151791161305840763762, 10.62002553710164858132584906439, 11.12944737388894344324574609371, 11.26174674038313996011824309538, 11.82621272311692655022977411977, 12.49280519255927801192241926355, 12.89953516361972655082105256017, 13.67849113108650617955798322613, 13.73658811430443438422004896969, 15.01197277949950273024189208868
