| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 5·11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s − 3·19-s + 20-s − 21-s + 5·22-s + 3·23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.218·21-s + 1.06·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.391327956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.391327956\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−12.43453437952784, −12.06764938031373, −11.60501356932993, −11.26928058144265, −10.89577227816309, −10.22106803409942, −9.754823737550743, −9.613698400182120, −8.729474897478115, −8.478225748546189, −7.831339844433307, −7.194302738278194, −6.859953580065063, −6.380527155421679, −5.981583664581080, −5.526956149318975, −4.894391717546041, −4.635363491520541, −4.047151923497092, −3.472347486297447, −3.048554157320832, −2.244463917358470, −1.649564715954706, −1.273887235204056, −0.5563772217121420,
0.5563772217121420, 1.273887235204056, 1.649564715954706, 2.244463917358470, 3.048554157320832, 3.472347486297447, 4.047151923497092, 4.635363491520541, 4.894391717546041, 5.526956149318975, 5.981583664581080, 6.380527155421679, 6.859953580065063, 7.194302738278194, 7.831339844433307, 8.478225748546189, 8.729474897478115, 9.613698400182120, 9.754823737550743, 10.22106803409942, 10.89577227816309, 11.26928058144265, 11.60501356932993, 12.06764938031373, 12.43453437952784
