| L(s) = 1 | − 2·4-s − 5-s − 7-s + 3·11-s + 13-s + 4·16-s + 3·17-s − 4·19-s + 2·20-s + 9·23-s + 25-s + 2·28-s + 6·29-s + 2·31-s + 35-s − 37-s + 3·41-s + 2·43-s − 6·44-s + 6·47-s − 6·49-s − 2·52-s − 9·53-s − 3·55-s + 12·59-s + 5·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s + 0.904·11-s + 0.277·13-s + 16-s + 0.727·17-s − 0.917·19-s + 0.447·20-s + 1.87·23-s + 1/5·25-s + 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.169·35-s − 0.164·37-s + 0.468·41-s + 0.304·43-s − 0.904·44-s + 0.875·47-s − 6/7·49-s − 0.277·52-s − 1.23·53-s − 0.404·55-s + 1.56·59-s + 0.640·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082399395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082399395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62510200039904674439234856658, −9.713611393939176921168495431920, −8.906263740943597793516749236173, −8.292416872807615906112471664419, −7.12261475792635686075443478244, −6.16181348679141975757729496775, −4.96599442706405448365610251513, −4.07858389118534067025312612810, −3.11322967677606911671732631614, −0.968476842474168100184639004266,
0.968476842474168100184639004266, 3.11322967677606911671732631614, 4.07858389118534067025312612810, 4.96599442706405448365610251513, 6.16181348679141975757729496775, 7.12261475792635686075443478244, 8.292416872807615906112471664419, 8.906263740943597793516749236173, 9.713611393939176921168495431920, 10.62510200039904674439234856658
