| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s − 1.23·5-s + 1.61·6-s − 4.23·7-s − 2.23·8-s + 9-s − 2.00·10-s + 4.23·11-s + 0.618·12-s + 3.47·13-s − 6.85·14-s − 1.23·15-s − 4.85·16-s + 3·17-s + 1.61·18-s − 2.76·19-s − 0.763·20-s − 4.23·21-s + 6.85·22-s + 5.70·23-s − 2.23·24-s − 3.47·25-s + 5.61·26-s + 27-s − 2.61·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.552·5-s + 0.660·6-s − 1.60·7-s − 0.790·8-s + 0.333·9-s − 0.632·10-s + 1.27·11-s + 0.178·12-s + 0.962·13-s − 1.83·14-s − 0.319·15-s − 1.21·16-s + 0.727·17-s + 0.381·18-s − 0.634·19-s − 0.170·20-s − 0.924·21-s + 1.46·22-s + 1.19·23-s − 0.456·24-s − 0.694·25-s + 1.10·26-s + 0.192·27-s − 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529140797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529140797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−14.13784562908028887463274991736, −13.08126410111092776195400260092, −12.54842031959081348843701926982, −11.31442079433776982465479980330, −9.594911363765538106327790540763, −8.787088278780598681714873590880, −6.96227678862324707950148129167, −5.92447139966814564047583105421, −3.96740643999368951175333635084, −3.33284344106940796359913592368,
3.33284344106940796359913592368, 3.96740643999368951175333635084, 5.92447139966814564047583105421, 6.96227678862324707950148129167, 8.787088278780598681714873590880, 9.594911363765538106327790540763, 11.31442079433776982465479980330, 12.54842031959081348843701926982, 13.08126410111092776195400260092, 14.13784562908028887463274991736
