| L(s) = 1 | − 2-s − 7·4-s − 5·5-s − 24·7-s + 15·8-s + 5·10-s − 52·11-s + 22·13-s + 24·14-s + 41·16-s + 14·17-s − 20·19-s + 35·20-s + 52·22-s + 168·23-s + 25·25-s − 22·26-s + 168·28-s − 230·29-s − 288·31-s − 161·32-s − 14·34-s + 120·35-s − 34·37-s + 20·38-s − 75·40-s − 122·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.447·5-s − 1.29·7-s + 0.662·8-s + 0.158·10-s − 1.42·11-s + 0.469·13-s + 0.458·14-s + 0.640·16-s + 0.199·17-s − 0.241·19-s + 0.391·20-s + 0.503·22-s + 1.52·23-s + 1/5·25-s − 0.165·26-s + 1.13·28-s − 1.47·29-s − 1.66·31-s − 0.889·32-s − 0.0706·34-s + 0.579·35-s − 0.151·37-s + 0.0853·38-s − 0.296·40-s − 0.464·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 122 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 338 T + p^{3} T^{2} \) |
| 59 | \( 1 + 100 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 - 328 T + p^{3} T^{2} \) |
| 73 | \( 1 + 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 + 330 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.88281880906420366216561430472, −13.17474530998481958083618292825, −12.90936309605119777396741352867, −10.92490624174672551127202315872, −9.784651796400427182871769727327, −8.651382954968274991345626734292, −7.27957773959223092455774425541, −5.35319961200548435385337242303, −3.48995313109668283817852366652, 0,
3.48995313109668283817852366652, 5.35319961200548435385337242303, 7.27957773959223092455774425541, 8.651382954968274991345626734292, 9.784651796400427182871769727327, 10.92490624174672551127202315872, 12.90936309605119777396741352867, 13.17474530998481958083618292825, 14.88281880906420366216561430472
