| L(s) = 1 | + 4.68·2-s − 3·3-s + 13.9·4-s − 15.0·5-s − 14.0·6-s + 33.3·7-s + 28.0·8-s + 9·9-s − 70.4·10-s − 11·11-s − 41.9·12-s + 4.73·13-s + 156.·14-s + 45.1·15-s + 19.5·16-s + 126.·17-s + 42.1·18-s + 91.3·19-s − 210.·20-s − 100.·21-s − 51.5·22-s − 136.·23-s − 84.0·24-s + 101.·25-s + 22.2·26-s − 27·27-s + 466.·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 0.577·3-s + 1.74·4-s − 1.34·5-s − 0.956·6-s + 1.80·7-s + 1.23·8-s + 0.333·9-s − 2.22·10-s − 0.301·11-s − 1.00·12-s + 0.101·13-s + 2.98·14-s + 0.776·15-s + 0.304·16-s + 1.81·17-s + 0.552·18-s + 1.10·19-s − 2.34·20-s − 1.04·21-s − 0.499·22-s − 1.23·23-s − 0.714·24-s + 0.809·25-s + 0.167·26-s − 0.192·27-s + 3.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.128709033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.128709033\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 4.68T + 8T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 7 | \( 1 - 33.3T + 343T^{2} \) |
| 13 | \( 1 - 4.73T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 26.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 307.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 244.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 218.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 682.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 889.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 907.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 379.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−8.339162490514265775626960724140, −7.63485903285689354271932133689, −7.38957362646280260536947255596, −5.99801114398747322606202937301, −5.32471546653120936205346446851, −4.82229462788075796633372314253, −3.97061283453390545462624777501, −3.41324453954198300141070756110, −2.03784954864004528335102238205, −0.861967137695153840557124121924,
0.861967137695153840557124121924, 2.03784954864004528335102238205, 3.41324453954198300141070756110, 3.97061283453390545462624777501, 4.82229462788075796633372314253, 5.32471546653120936205346446851, 5.99801114398747322606202937301, 7.38957362646280260536947255596, 7.63485903285689354271932133689, 8.339162490514265775626960724140
