| L(s) = 1 | + 2-s + 3.12·3-s + 4-s + 5-s + 3.12·6-s − 0.193·7-s + 8-s + 6.78·9-s + 10-s − 11-s + 3.12·12-s + 6.24·13-s − 0.193·14-s + 3.12·15-s + 16-s + 4.69·17-s + 6.78·18-s − 3.31·19-s + 20-s − 0.605·21-s − 22-s − 4.24·23-s + 3.12·24-s + 25-s + 6.24·26-s + 11.8·27-s − 0.193·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.80·3-s + 0.5·4-s + 0.447·5-s + 1.27·6-s − 0.0731·7-s + 0.353·8-s + 2.26·9-s + 0.316·10-s − 0.301·11-s + 0.903·12-s + 1.73·13-s − 0.0517·14-s + 0.807·15-s + 0.250·16-s + 1.13·17-s + 1.59·18-s − 0.761·19-s + 0.223·20-s − 0.132·21-s − 0.213·22-s − 0.886·23-s + 0.638·24-s + 0.200·25-s + 1.22·26-s + 2.28·27-s − 0.0365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.130467673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.130467673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 7 | \( 1 + 0.193T + 7T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 + 3.31T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 6.26T + 41T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.75T + 71T^{2} \) |
| 73 | \( 1 + 5.20T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 + 5.37T + 89T^{2} \) |
| 97 | \( 1 + 2.37T + 97T^{2} \) |
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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
−8.467551501846777607759061898799, −7.61861530705069056726716806354, −6.96289758480569140135735006426, −6.06020433320541590949023286585, −5.38096135520647808054760734206, −4.19242288012464573151339029146, −3.62375168379732252714482659430, −3.10075796695319383267878775483, −2.05726273503079414713591679882, −1.47374920379272516696865367168,
1.47374920379272516696865367168, 2.05726273503079414713591679882, 3.10075796695319383267878775483, 3.62375168379732252714482659430, 4.19242288012464573151339029146, 5.38096135520647808054760734206, 6.06020433320541590949023286585, 6.96289758480569140135735006426, 7.61861530705069056726716806354, 8.467551501846777607759061898799
