| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 5-s − 7-s + 2.23·8-s − 1.61·10-s + 4.23·11-s + 1.38·13-s + 1.61·14-s − 4.85·16-s + 1.61·17-s − 7.09·19-s + 0.618·20-s − 6.85·22-s + 5.38·23-s + 25-s − 2.23·26-s − 0.618·28-s − 9.56·29-s + 6.70·31-s + 3.38·32-s − 2.61·34-s − 35-s + 6.70·37-s + 11.4·38-s + 2.23·40-s + 8.09·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.447·5-s − 0.377·7-s + 0.790·8-s − 0.511·10-s + 1.27·11-s + 0.383·13-s + 0.432·14-s − 1.21·16-s + 0.392·17-s − 1.62·19-s + 0.138·20-s − 1.46·22-s + 1.12·23-s + 0.200·25-s − 0.438·26-s − 0.116·28-s − 1.77·29-s + 1.20·31-s + 0.597·32-s − 0.448·34-s − 0.169·35-s + 1.10·37-s + 1.86·38-s + 0.353·40-s + 1.26·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9086859901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9086859901\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 2T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 9.56T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 - 11T + 47T^{2} \) |
| 53 | \( 1 + 4.38T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−9.753766866729512763841275966251, −9.248483680802985052533222341712, −8.640742040769503528056593481942, −7.70515622230534988845025595180, −6.71727343522141888347397982032, −6.06620526418433562254995509608, −4.68200202231709713467618930011, −3.70585483573006111685721921414, −2.13571733821371376763959408890, −0.935176857455447708880149400432,
0.935176857455447708880149400432, 2.13571733821371376763959408890, 3.70585483573006111685721921414, 4.68200202231709713467618930011, 6.06620526418433562254995509608, 6.71727343522141888347397982032, 7.70515622230534988845025595180, 8.640742040769503528056593481942, 9.248483680802985052533222341712, 9.753766866729512763841275966251
