| L(s) = 1 | + 2.23·2-s + 3-s + 3.00·4-s + 2.23·6-s − 3.23·7-s + 2.23·8-s + 9-s + 4·11-s + 3.00·12-s + 4.47·13-s − 7.23·14-s − 0.999·16-s + 2.76·17-s + 2.23·18-s + 7.23·19-s − 3.23·21-s + 8.94·22-s − 23-s + 2.23·24-s + 10.0·26-s + 27-s − 9.70·28-s + 4.47·29-s − 6.47·31-s − 6.70·32-s + 4·33-s + 6.18·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.912·6-s − 1.22·7-s + 0.790·8-s + 0.333·9-s + 1.20·11-s + 0.866·12-s + 1.24·13-s − 1.93·14-s − 0.249·16-s + 0.670·17-s + 0.527·18-s + 1.66·19-s − 0.706·21-s + 1.90·22-s − 0.208·23-s + 0.456·24-s + 1.96·26-s + 0.192·27-s − 1.83·28-s + 0.830·29-s − 1.16·31-s − 1.18·32-s + 0.696·33-s + 1.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.109993219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.109993219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−9.299069256035598976488010760525, −8.653978763543362151115276256575, −7.35815629138189633660207703826, −6.69711758405768371451628293532, −6.00492176062609971528052945382, −5.25872329857534446272745279929, −3.96995804489005579109246465406, −3.54131370540973753219953631174, −2.91853978046178974766531513229, −1.40586227760554875656016455010,
1.40586227760554875656016455010, 2.91853978046178974766531513229, 3.54131370540973753219953631174, 3.96995804489005579109246465406, 5.25872329857534446272745279929, 6.00492176062609971528052945382, 6.69711758405768371451628293532, 7.35815629138189633660207703826, 8.653978763543362151115276256575, 9.299069256035598976488010760525
