| L(s) = 1 | + 2-s − 2·4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s + 2·11-s − 4·13-s − 2·14-s + 16-s − 4·17-s − 6·19-s + 4·20-s + 2·22-s + 8·23-s + 3·25-s − 4·26-s + 4·28-s + 2·29-s − 18·31-s + 2·32-s − 4·34-s + 4·35-s − 6·37-s − 6·38-s + 6·40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s + 0.603·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.894·20-s + 0.426·22-s + 1.66·23-s + 3/5·25-s − 0.784·26-s + 0.755·28-s + 0.371·29-s − 3.23·31-s + 0.353·32-s − 0.685·34-s + 0.676·35-s − 0.986·37-s − 0.973·38-s + 0.948·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 69 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.556106386339217141460464027898, −9.391706622678198002854208922000, −8.967960470462966836472835956708, −8.729898990427414556258071285606, −8.376995478464307025008966230424, −7.60898122080900650959498222077, −7.10348132681085543433714852055, −6.98856901807420140047761268445, −6.45992653265138880894882843728, −5.84907564284921514787621089493, −5.15274773476610309384399417678, −5.05497557256079244629396029482, −4.31363723041279263630641516682, −4.20756096555383868349042482036, −3.38679108748599707894809493918, −3.34529236639540673590233895649, −2.40982634328725629809070073615, −1.59538412249758020081216197770, 0, 0,
1.59538412249758020081216197770, 2.40982634328725629809070073615, 3.34529236639540673590233895649, 3.38679108748599707894809493918, 4.20756096555383868349042482036, 4.31363723041279263630641516682, 5.05497557256079244629396029482, 5.15274773476610309384399417678, 5.84907564284921514787621089493, 6.45992653265138880894882843728, 6.98856901807420140047761268445, 7.10348132681085543433714852055, 7.60898122080900650959498222077, 8.376995478464307025008966230424, 8.729898990427414556258071285606, 8.967960470462966836472835956708, 9.391706622678198002854208922000, 9.556106386339217141460464027898
