| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 5-s + 7-s + 2.23·8-s + 1.61·10-s − 11-s − 5.47·13-s − 1.61·14-s − 4.85·16-s − 0.763·17-s + 6.70·19-s − 0.618·20-s + 1.61·22-s + 7.70·23-s − 4·25-s + 8.85·26-s + 0.618·28-s − 5·29-s − 0.763·31-s + 3.38·32-s + 1.23·34-s − 35-s − 7·37-s − 10.8·38-s − 2.23·40-s − 6.47·41-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 − 0.301·11-s − 1.51·13-s − 0.432·14-s − 1.21·16-s − 0.185·17-s + 1.53·19-s − 0.138·20-s + 0.344·22-s + 1.60·23-s − 0.800·25-s + 1.73·26-s + 0.116·28-s − 0.928·29-s − 0.137·31-s + 0.597·32-s + 0.211·34-s − 0.169·35-s − 1.15·37-s − 1.76·38-s − 0.353·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 0.763T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 5.52T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.814424875318201484504423045261, −9.274956371802362232284804134023, −8.325634047048986700067163117070, −7.45565973461179382911003240239, −7.12213219192332975706190507539, −5.32716379192239946664751496737, −4.64794773018248815477062693086, −3.15367387378703832061078611773, −1.64844489523905234411263309100, 0,
1.64844489523905234411263309100, 3.15367387378703832061078611773, 4.64794773018248815477062693086, 5.32716379192239946664751496737, 7.12213219192332975706190507539, 7.45565973461179382911003240239, 8.325634047048986700067163117070, 9.274956371802362232284804134023, 9.814424875318201484504423045261
