| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 2.61·5-s + 3·7-s − 2.23·8-s + 1.61·10-s + 0.763·11-s + 4.85·13-s + 1.85·14-s + 1.85·16-s − 0.236·17-s + 5·19-s − 4.23·20-s + 0.472·22-s + 5.47·23-s + 1.85·25-s + 3.00·26-s − 4.85·28-s + 8.61·29-s + 5.61·32-s − 0.145·34-s + 7.85·35-s + 0.236·37-s + 3.09·38-s − 5.85·40-s − 6.47·41-s − 4.61·43-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 1.17·5-s + 1.13·7-s − 0.790·8-s + 0.511·10-s + 0.230·11-s + 1.34·13-s + 0.495·14-s + 0.463·16-s − 0.0572·17-s + 1.14·19-s − 0.947·20-s + 0.100·22-s + 1.14·23-s + 0.370·25-s + 0.588·26-s − 0.917·28-s + 1.60·29-s + 0.993·32-s − 0.0250·34-s + 1.32·35-s + 0.0388·37-s + 0.501·38-s − 0.925·40-s − 1.01·41-s − 0.704·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.762248334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.762248334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 - 3.38T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 0.0901T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−7.924259286070155661360703687216, −6.93557529148765512587656196269, −6.17190875026419003414481329808, −5.58650080961782350111947535772, −5.01392599008245585185330465269, −4.45385253036367007764708372240, −3.50246521746916502767250162303, −2.78120328883169566648269881702, −1.55272497947463218213950297901, −1.02688990296808773455271997397,
1.02688990296808773455271997397, 1.55272497947463218213950297901, 2.78120328883169566648269881702, 3.50246521746916502767250162303, 4.45385253036367007764708372240, 5.01392599008245585185330465269, 5.58650080961782350111947535772, 6.17190875026419003414481329808, 6.93557529148765512587656196269, 7.924259286070155661360703687216
