| L(s) = 1 | − 2.30·2-s + 3.30·4-s − 5-s + 7-s − 3.00·8-s + 2.30·10-s − 3·11-s + 1.30·13-s − 2.30·14-s + 0.302·16-s + 2.30·17-s − 3.30·19-s − 3.30·20-s + 6.90·22-s − 0.697·23-s + 25-s − 3·26-s + 3.30·28-s − 5.30·29-s − 2.39·31-s + 5.30·32-s − 5.30·34-s − 35-s + 3.60·37-s + 7.60·38-s + 3.00·40-s + 6.90·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s + 0.728·10-s − 0.904·11-s + 0.361·13-s − 0.615·14-s + 0.0756·16-s + 0.558·17-s − 0.757·19-s − 0.738·20-s + 1.47·22-s − 0.145·23-s + 0.200·25-s − 0.588·26-s + 0.624·28-s − 0.984·29-s − 0.430·31-s + 0.937·32-s − 0.909·34-s − 0.169·35-s + 0.592·37-s + 1.23·38-s + 0.474·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 23 | \( 1 + 0.697T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.09T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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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
−9.555203656778963903074831960031, −8.753108903507971250433795228050, −7.969054625215624680121296890813, −7.58391826096975298495845965602, −6.54542232984902087688770967563, −5.42865668409504728035112464771, −4.16183705672524791240038023341, −2.73692037708999870764562246318, −1.51779104103791489479495855848, 0,
1.51779104103791489479495855848, 2.73692037708999870764562246318, 4.16183705672524791240038023341, 5.42865668409504728035112464771, 6.54542232984902087688770967563, 7.58391826096975298495845965602, 7.969054625215624680121296890813, 8.753108903507971250433795228050, 9.555203656778963903074831960031
