| L(s) = 1 | − 2-s + 4-s + 2·5-s − 4·7-s − 8-s − 2·10-s + 11-s − 4·13-s + 4·14-s + 16-s + 17-s − 8·19-s + 2·20-s − 22-s − 25-s + 4·26-s − 4·28-s + 10·31-s − 32-s − 34-s − 8·35-s + 8·37-s + 8·38-s − 2·40-s + 10·41-s − 8·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s − 0.353·8-s − 0.632·10-s + 0.301·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 1.83·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 0.784·26-s − 0.755·28-s + 1.79·31-s − 0.176·32-s − 0.171·34-s − 1.35·35-s + 1.31·37-s + 1.29·38-s − 0.316·40-s + 1.56·41-s − 1.21·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005919273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005919273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.732171890413360673116508117085, −7.990157865470902753936524190848, −6.99198663690775524240303916195, −6.35899333986310493133674784342, −6.03464821741840975166082717516, −4.84564818542032239874263552235, −3.80499868903726682009410489312, −2.67833100802737035655116658967, −2.16018443002583249133409236496, −0.63145901629488314235509347358,
0.63145901629488314235509347358, 2.16018443002583249133409236496, 2.67833100802737035655116658967, 3.80499868903726682009410489312, 4.84564818542032239874263552235, 6.03464821741840975166082717516, 6.35899333986310493133674784342, 6.99198663690775524240303916195, 7.990157865470902753936524190848, 8.732171890413360673116508117085
