| L(s) = 1 | − 2.95·2-s − 23.2·4-s − 64.8·5-s − 160.·7-s + 163.·8-s + 191.·10-s + 206.·11-s − 65.7·13-s + 473.·14-s + 262.·16-s + 601.·17-s − 2.00e3·19-s + 1.50e3·20-s − 609.·22-s + 4.36e3·23-s + 1.08e3·25-s + 194.·26-s + 3.72e3·28-s + 934.·29-s + 5.25e3·31-s − 6.00e3·32-s − 1.77e3·34-s + 1.03e4·35-s + 5.68e3·37-s + 5.91e3·38-s − 1.05e4·40-s + 6.44e3·41-s + ⋯ |
| L(s) = 1 | − 0.522·2-s − 0.727·4-s − 1.16·5-s − 1.23·7-s + 0.901·8-s + 0.605·10-s + 0.514·11-s − 0.107·13-s + 0.645·14-s + 0.256·16-s + 0.504·17-s − 1.27·19-s + 0.843·20-s − 0.268·22-s + 1.72·23-s + 0.345·25-s + 0.0563·26-s + 0.898·28-s + 0.206·29-s + 0.981·31-s − 1.03·32-s − 0.263·34-s + 1.43·35-s + 0.682·37-s + 0.664·38-s − 1.04·40-s + 0.598·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5840509827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5840509827\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.95T + 32T^{2} \) |
| 5 | \( 1 + 64.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 206.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 65.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 601.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 934.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.81e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.35e4T + 8.58e9T^{2} \) |
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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
−13.16910211913300131824401916142, −12.42958028367456118707650436541, −11.07672424817775170751142692805, −9.857253644250702071694358371824, −8.897009295051039125021416536729, −7.81928360554241827883556204854, −6.53665570613410945791263685390, −4.57841166321421870250052315753, −3.38462259905508809332924363796, −0.61970476363678776205800583398,
0.61970476363678776205800583398, 3.38462259905508809332924363796, 4.57841166321421870250052315753, 6.53665570613410945791263685390, 7.81928360554241827883556204854, 8.897009295051039125021416536729, 9.857253644250702071694358371824, 11.07672424817775170751142692805, 12.42958028367456118707650436541, 13.16910211913300131824401916142
