| L(s) = 1 | − 2.58·2-s − 2.87·3-s + 4.68·4-s − 3.69·5-s + 7.43·6-s − 0.298·7-s − 6.95·8-s + 5.26·9-s + 9.56·10-s − 1.58·11-s − 13.4·12-s − 4.24·13-s + 0.771·14-s + 10.6·15-s + 8.61·16-s + 4.97·17-s − 13.6·18-s + 8.32·19-s − 17.3·20-s + 0.857·21-s + 4.09·22-s − 23-s + 20.0·24-s + 8.67·25-s + 10.9·26-s − 6.52·27-s − 1.39·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.66·3-s + 2.34·4-s − 1.65·5-s + 3.03·6-s − 0.112·7-s − 2.45·8-s + 1.75·9-s + 3.02·10-s − 0.477·11-s − 3.89·12-s − 1.17·13-s + 0.206·14-s + 2.74·15-s + 2.15·16-s + 1.20·17-s − 3.21·18-s + 1.90·19-s − 3.87·20-s + 0.187·21-s + 0.873·22-s − 0.208·23-s + 4.08·24-s + 1.73·25-s + 2.15·26-s − 1.25·27-s − 0.264·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 0.298T + 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 8.32T + 19T^{2} \) |
| 31 | \( 1 + 0.692T + 31T^{2} \) |
| 37 | \( 1 - 7.77T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 0.941T + 47T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 8.46T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 6.78T + 89T^{2} \) |
| 97 | \( 1 - 9.10T + 97T^{2} \) |
| 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
−9.996500084915148246881844337460, −9.567754348352624146425253215807, −8.008583712267892591754977310925, −7.58690633226087232054931087305, −7.00407423324327812750388531412, −5.79219354834585084886262317910, −4.75121756589594981719428905507, −3.12163582341770630101730797076, −1.01067586470011474398636321036, 0,
1.01067586470011474398636321036, 3.12163582341770630101730797076, 4.75121756589594981719428905507, 5.79219354834585084886262317910, 7.00407423324327812750388531412, 7.58690633226087232054931087305, 8.008583712267892591754977310925, 9.567754348352624146425253215807, 9.996500084915148246881844337460
