| L(s) = 1 | − 2.35·2-s − 3-s + 3.55·4-s − 5-s + 2.35·6-s + 1.32·7-s − 3.66·8-s + 9-s + 2.35·10-s + 3.45·11-s − 3.55·12-s − 13-s − 3.11·14-s + 15-s + 1.53·16-s + 0.209·17-s − 2.35·18-s − 7.21·19-s − 3.55·20-s − 1.32·21-s − 8.15·22-s + 3.54·23-s + 3.66·24-s + 25-s + 2.35·26-s − 27-s + 4.70·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.77·4-s − 0.447·5-s + 0.962·6-s + 0.500·7-s − 1.29·8-s + 0.333·9-s + 0.745·10-s + 1.04·11-s − 1.02·12-s − 0.277·13-s − 0.833·14-s + 0.258·15-s + 0.383·16-s + 0.0507·17-s − 0.555·18-s − 1.65·19-s − 0.795·20-s − 0.288·21-s − 1.73·22-s + 0.738·23-s + 0.748·24-s + 0.200·25-s + 0.462·26-s − 0.192·27-s + 0.889·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 3.45T + 11T^{2} \) |
| 17 | \( 1 - 0.209T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 0.413T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 2.57T + 71T^{2} \) |
| 73 | \( 1 - 6.17T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.316T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 + 7.84T + 97T^{2} \) |
| 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
−7.83209527386705688784163125008, −7.20090659505816503704936811822, −6.51758104246089471086289316423, −5.99226870072709998036856488322, −4.67887989776293019086698688361, −4.22618940230940313166376667389, −2.90126645313844804015724857762, −1.84278635808694810207186340298, −1.07540031195674656655791224506, 0,
1.07540031195674656655791224506, 1.84278635808694810207186340298, 2.90126645313844804015724857762, 4.22618940230940313166376667389, 4.67887989776293019086698688361, 5.99226870072709998036856488322, 6.51758104246089471086289316423, 7.20090659505816503704936811822, 7.83209527386705688784163125008
