| L(s) = 1 | + 2-s + 4-s − 4·5-s − 7-s + 8-s − 4·10-s + 11-s + 13-s − 14-s + 16-s − 4·17-s + 2·19-s − 4·20-s + 22-s + 7·23-s + 11·25-s + 26-s − 28-s + 8·29-s + 3·31-s + 32-s − 4·34-s + 4·35-s + 7·37-s + 2·38-s − 4·40-s + 7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s + 0.538·31-s + 0.176·32-s − 0.685·34-s + 0.676·35-s + 1.15·37-s + 0.324·38-s − 0.632·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839748737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839748737\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.229708445181394321480923526929, −8.446510046834741944724142023215, −7.70223365400781146351389504873, −6.89743886981333030203070546273, −6.32586188918121275548015521553, −4.91521481773069545833368645820, −4.38408648015283815906866115610, −3.48622278921391586115492321920, −2.77045286705004577101314310504, −0.859680047349367001421384183810,
0.859680047349367001421384183810, 2.77045286705004577101314310504, 3.48622278921391586115492321920, 4.38408648015283815906866115610, 4.91521481773069545833368645820, 6.32586188918121275548015521553, 6.89743886981333030203070546273, 7.70223365400781146351389504873, 8.446510046834741944724142023215, 9.229708445181394321480923526929
