| L(s) = 1 | + 2-s − 3.16·3-s + 4-s − 3.16·6-s + 7-s + 8-s + 7.00·9-s − 11-s − 3.16·12-s + 6.16·13-s + 14-s + 16-s + 7.16·17-s + 7.00·18-s − 19-s − 3.16·21-s − 22-s + 4.16·23-s − 3.16·24-s + 6.16·26-s − 12.6·27-s + 28-s + 3·29-s − 7·31-s + 32-s + 3.16·33-s + 7.16·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.82·3-s + 0.5·4-s − 1.29·6-s + 0.377·7-s + 0.353·8-s + 2.33·9-s − 0.301·11-s − 0.912·12-s + 1.70·13-s + 0.267·14-s + 0.250·16-s + 1.73·17-s + 1.64·18-s − 0.229·19-s − 0.690·21-s − 0.213·22-s + 0.867·23-s − 0.645·24-s + 1.20·26-s − 2.43·27-s + 0.188·28-s + 0.557·29-s − 1.25·31-s + 0.176·32-s + 0.550·33-s + 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995525542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995525542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 13 | \( 1 - 6.16T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 + 2.32T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 0.324T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 4.16T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
| 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
−8.272292759384483791429275543328, −7.48334304928817553177256800987, −6.69411201315076743408900183037, −6.09841073135851032696648786407, −5.38355978354408380530155307473, −5.07896506943886261362105384021, −4.01735095556557236005539977597, −3.32864513428137456807567405196, −1.65877905827825118838421535640, −0.880963139309498179426257595235,
0.880963139309498179426257595235, 1.65877905827825118838421535640, 3.32864513428137456807567405196, 4.01735095556557236005539977597, 5.07896506943886261362105384021, 5.38355978354408380530155307473, 6.09841073135851032696648786407, 6.69411201315076743408900183037, 7.48334304928817553177256800987, 8.272292759384483791429275543328
