L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3.63·7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 1.63·13-s + 3.63·14-s − 15-s + 16-s + 17-s + 18-s + 7.63·19-s − 20-s + 3.63·21-s − 22-s − 0.361·23-s + 24-s + 25-s − 1.63·26-s + 27-s + 3.63·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.37·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.454·13-s + 0.972·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.75·19-s − 0.223·20-s + 0.794·21-s − 0.213·22-s − 0.0752·23-s + 0.204·24-s + 0.200·25-s − 0.321·26-s + 0.192·27-s + 0.687·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.555784007\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.555784007\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 3.63T + 7T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 + 0.361T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8.52T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 2.88T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.39T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 1.20T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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
−8.009116130616066712057018635225, −7.35941477554438430358384830071, −7.06088387201508506871747512798, −5.55985628675021215554984245715, −5.31287241451343350324195641174, −4.42036035790833188444687031653, −3.75784329859456685803801349991, −2.89048129332289778999117090659, −2.05030075191844266794782363152, −1.06384498742223717873739093744,
1.06384498742223717873739093744, 2.05030075191844266794782363152, 2.89048129332289778999117090659, 3.75784329859456685803801349991, 4.42036035790833188444687031653, 5.31287241451343350324195641174, 5.55985628675021215554984245715, 7.06088387201508506871747512798, 7.35941477554438430358384830071, 8.009116130616066712057018635225