| L(s) = 1 | − 1.30·2-s − 0.302·4-s + 5-s − 7-s + 3·8-s − 1.30·10-s + 11-s − 3.60·13-s + 1.30·14-s − 3.30·16-s + 4·17-s + 3·19-s − 0.302·20-s − 1.30·22-s + 2·23-s − 4·25-s + 4.69·26-s + 0.302·28-s + 1.60·29-s − 2·31-s − 1.69·32-s − 5.21·34-s − 35-s + 6.21·37-s − 3.90·38-s + 3·40-s + 7.21·41-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.411·10-s + 0.301·11-s − 1.00·13-s + 0.348·14-s − 0.825·16-s + 0.970·17-s + 0.688·19-s − 0.0677·20-s − 0.277·22-s + 0.417·23-s − 0.800·25-s + 0.921·26-s + 0.0572·28-s + 0.298·29-s − 0.359·31-s − 0.300·32-s − 0.893·34-s − 0.169·35-s + 1.02·37-s − 0.634·38-s + 0.474·40-s + 1.12·41-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
| \(L(1)\) |
\(\approx\) |
\(0.8605196024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8605196024\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 9.21T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 0.394T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 + 3.21T + 71T^{2} \) |
| 73 | \( 1 - 8.81T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
| 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
−10.13463334997917319053830199274, −9.535321875702247202951795485341, −9.055841113559769311009738696182, −7.78842653525444615173035388902, −7.34586334390177520442197835540, −6.03889810800677308375009024200, −5.08014095816749924744764638955, −3.91970587248471631200002206855, −2.45518365772099446163396327805, −0.933037383348076620158084828454,
0.933037383348076620158084828454, 2.45518365772099446163396327805, 3.91970587248471631200002206855, 5.08014095816749924744764638955, 6.03889810800677308375009024200, 7.34586334390177520442197835540, 7.78842653525444615173035388902, 9.055841113559769311009738696182, 9.535321875702247202951795485341, 10.13463334997917319053830199274
