| L(s) = 1 | + 2.25·2-s + 3·3-s − 2.91·4-s − 0.928·5-s + 6.76·6-s − 11.4·7-s − 24.6·8-s + 9·9-s − 2.09·10-s + 11·11-s − 8.75·12-s − 58.0·13-s − 25.7·14-s − 2.78·15-s − 32.1·16-s − 123.·17-s + 20.2·18-s + 150.·19-s + 2.71·20-s − 34.2·21-s + 24.7·22-s − 67.5·23-s − 73.8·24-s − 124.·25-s − 130.·26-s + 27·27-s + 33.2·28-s + ⋯ |
| L(s) = 1 | + 0.797·2-s + 0.577·3-s − 0.364·4-s − 0.0830·5-s + 0.460·6-s − 0.615·7-s − 1.08·8-s + 0.333·9-s − 0.0662·10-s + 0.301·11-s − 0.210·12-s − 1.23·13-s − 0.490·14-s − 0.0479·15-s − 0.502·16-s − 1.75·17-s + 0.265·18-s + 1.82·19-s + 0.0303·20-s − 0.355·21-s + 0.240·22-s − 0.612·23-s − 0.628·24-s − 0.993·25-s − 0.987·26-s + 0.192·27-s + 0.224·28-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
| \(L(2)\) |
\(\approx\) |
\(2.195615240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.195615240\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.25T + 8T^{2} \) |
| 5 | \( 1 + 0.928T + 125T^{2} \) |
| 7 | \( 1 + 11.4T + 343T^{2} \) |
| 13 | \( 1 + 58.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 47.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 416.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 203.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 11.6T + 2.05e5T^{2} \) |
| 67 | \( 1 + 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.12T + 3.89e5T^{2} \) |
| 79 | \( 1 + 448.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 533.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 663.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 999.T + 9.12e5T^{2} \) |
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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
−9.056268672924613426536082937254, −7.922415672423034012609418023922, −7.23934998242458964115837331229, −6.30187264882104750278655698861, −5.51103358509344277774394382929, −4.52080132599914476331120923572, −3.98564450476525133539107836686, −2.98941189830540557760398762043, −2.26629294051865824735781518468, −0.56184999844669341609283823021,
0.56184999844669341609283823021, 2.26629294051865824735781518468, 2.98941189830540557760398762043, 3.98564450476525133539107836686, 4.52080132599914476331120923572, 5.51103358509344277774394382929, 6.30187264882104750278655698861, 7.23934998242458964115837331229, 7.922415672423034012609418023922, 9.056268672924613426536082937254
