| L(s) = 1 | + 2-s + 3-s + 4-s − 3.23·5-s + 6-s + 2·7-s + 8-s + 9-s − 3.23·10-s − 11-s + 12-s + 0.763·13-s + 2·14-s − 3.23·15-s + 16-s + 4.47·17-s + 18-s + 19-s − 3.23·20-s + 2·21-s − 22-s + 7.70·23-s + 24-s + 5.47·25-s + 0.763·26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.44·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s − 0.301·11-s + 0.288·12-s + 0.211·13-s + 0.534·14-s − 0.835·15-s + 0.250·16-s + 1.08·17-s + 0.235·18-s + 0.229·19-s − 0.723·20-s + 0.436·21-s − 0.213·22-s + 1.60·23-s + 0.204·24-s + 1.09·25-s + 0.149·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.812051325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.812051325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.728869631079103147539874899268, −8.540553645059729523992104427175, −7.961979839717142258750964307095, −7.44106196520399250966194683031, −6.46500892459673180295756569395, −5.09985948165748119112158820363, −4.54651272707773043293834808348, −3.51536257709699716727703330746, −2.85717303269076822508916571979, −1.20462983542259946438258617303,
1.20462983542259946438258617303, 2.85717303269076822508916571979, 3.51536257709699716727703330746, 4.54651272707773043293834808348, 5.09985948165748119112158820363, 6.46500892459673180295756569395, 7.44106196520399250966194683031, 7.961979839717142258750964307095, 8.540553645059729523992104427175, 9.728869631079103147539874899268
