| L(s) = 1 | − 1.24·2-s − 3-s + 0.554·4-s − 5-s + 1.24·6-s + 0.554·8-s + 9-s + 1.24·10-s − 0.554·12-s + 15-s − 1.24·16-s + 1.80·17-s − 1.24·18-s − 0.445·19-s − 0.554·20-s − 1.24·23-s − 0.554·24-s + 25-s − 27-s − 1.24·30-s − 1.80·31-s + 0.999·32-s − 2.24·34-s + 0.554·36-s + 0.554·38-s − 0.554·40-s − 45-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 3-s + 0.554·4-s − 5-s + 1.24·6-s + 0.554·8-s + 9-s + 1.24·10-s − 0.554·12-s + 15-s − 1.24·16-s + 1.80·17-s − 1.24·18-s − 0.445·19-s − 0.554·20-s − 1.24·23-s − 0.554·24-s + 25-s − 27-s − 1.24·30-s − 1.80·31-s + 0.999·32-s − 2.24·34-s + 0.554·36-s + 0.554·38-s − 0.554·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3054950224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3054950224\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.80T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 + 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.445T + T^{2} \) |
| 53 | \( 1 - 0.445T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - 1.80T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.159067924565842081809914685048, −8.218080873819125555240119072039, −7.59899322702754890090638907158, −7.18965014798507772565875998171, −6.10381402575874155158716425644, −5.26386849660629311311194633307, −4.30835199473103575748946301270, −3.54889004477366176270879392944, −1.83042470538908035507727379061, −0.66677086254804536521175146278,
0.66677086254804536521175146278, 1.83042470538908035507727379061, 3.54889004477366176270879392944, 4.30835199473103575748946301270, 5.26386849660629311311194633307, 6.10381402575874155158716425644, 7.18965014798507772565875998171, 7.59899322702754890090638907158, 8.218080873819125555240119072039, 9.159067924565842081809914685048
