| L(s) = 1 | − 3-s − 5-s − 0.395·7-s + 9-s − 2.79·11-s + 5.70·13-s + 15-s + 4.25·17-s + 6.10·19-s + 0.395·21-s + 6.36·23-s + 25-s − 27-s + 7.96·29-s − 31-s + 2.79·33-s + 0.395·35-s − 9.80·37-s − 5.70·39-s + 3.20·41-s + 3.31·43-s − 45-s − 7.05·47-s − 6.84·49-s − 4.25·51-s − 1.05·53-s + 2.79·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.149·7-s + 0.333·9-s − 0.841·11-s + 1.58·13-s + 0.258·15-s + 1.03·17-s + 1.40·19-s + 0.0863·21-s + 1.32·23-s + 0.200·25-s − 0.192·27-s + 1.47·29-s − 0.179·31-s + 0.486·33-s + 0.0669·35-s − 1.61·37-s − 0.913·39-s + 0.501·41-s + 0.504·43-s − 0.149·45-s − 1.02·47-s − 0.977·49-s − 0.596·51-s − 0.144·53-s + 0.376·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680143867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680143867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.395T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 3.20T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 0.156T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 - 7.55T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−7.908444729810245126205362277391, −7.14743018970984496659977866312, −6.55156689910090769139081628889, −5.65960946002533526550397483843, −5.22617414292820246563334613942, −4.42558099419097340603814272072, −3.32711579910307443639748951286, −3.07478615530304964810777717843, −1.47719596001949104158178328434, −0.73778774823554360209077541665,
0.73778774823554360209077541665, 1.47719596001949104158178328434, 3.07478615530304964810777717843, 3.32711579910307443639748951286, 4.42558099419097340603814272072, 5.22617414292820246563334613942, 5.65960946002533526550397483843, 6.55156689910090769139081628889, 7.14743018970984496659977866312, 7.908444729810245126205362277391
