| L(s) = 1 | − 1.81·2-s + 3-s + 1.28·4-s + 1.68·5-s − 1.81·6-s + 2.36·7-s + 1.29·8-s + 9-s − 3.06·10-s − 6.20·11-s + 1.28·12-s − 0.435·13-s − 4.28·14-s + 1.68·15-s − 4.91·16-s − 17-s − 1.81·18-s + 3.20·19-s + 2.17·20-s + 2.36·21-s + 11.2·22-s + 7.21·23-s + 1.29·24-s − 2.14·25-s + 0.789·26-s + 27-s + 3.03·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.643·4-s + 0.755·5-s − 0.740·6-s + 0.892·7-s + 0.457·8-s + 0.333·9-s − 0.968·10-s − 1.87·11-s + 0.371·12-s − 0.120·13-s − 1.14·14-s + 0.436·15-s − 1.22·16-s − 0.242·17-s − 0.427·18-s + 0.736·19-s + 0.486·20-s + 0.515·21-s + 2.39·22-s + 1.50·23-s + 0.263·24-s − 0.429·25-s + 0.154·26-s + 0.192·27-s + 0.574·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.439850146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439850146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 0.435T + 13T^{2} \) |
| 19 | \( 1 - 3.20T + 19T^{2} \) |
| 23 | \( 1 - 7.21T + 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 + 2.98T + 37T^{2} \) |
| 41 | \( 1 - 8.05T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.902T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
| 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
−7.80836472369647136072409846740, −7.61170020726385612495333808227, −6.82341587992234227411006511649, −5.67997731582294452554866966910, −5.06533028931359206888732433134, −4.47564969393563779637418156980, −3.12317362242659502764845257064, −2.36036183999696944999589529047, −1.73132036218047591541469531443, −0.71282426099098889362713569105,
0.71282426099098889362713569105, 1.73132036218047591541469531443, 2.36036183999696944999589529047, 3.12317362242659502764845257064, 4.47564969393563779637418156980, 5.06533028931359206888732433134, 5.67997731582294452554866966910, 6.82341587992234227411006511649, 7.61170020726385612495333808227, 7.80836472369647136072409846740
