| L(s) = 1 | − 3.85·2-s + 6.86·4-s − 5.82·5-s − 7·7-s + 4.38·8-s + 22.4·10-s − 11·11-s − 87.1·13-s + 26.9·14-s − 71.8·16-s + 119.·17-s + 23.9·19-s − 39.9·20-s + 42.4·22-s − 119.·23-s − 91.0·25-s + 335.·26-s − 48.0·28-s + 53.7·29-s − 69.0·31-s + 241.·32-s − 458.·34-s + 40.7·35-s + 28.6·37-s − 92.4·38-s − 25.5·40-s − 419.·41-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.857·4-s − 0.521·5-s − 0.377·7-s + 0.193·8-s + 0.710·10-s − 0.301·11-s − 1.85·13-s + 0.515·14-s − 1.12·16-s + 1.69·17-s + 0.289·19-s − 0.447·20-s + 0.410·22-s − 1.08·23-s − 0.728·25-s + 2.53·26-s − 0.324·28-s + 0.344·29-s − 0.399·31-s + 1.33·32-s − 2.31·34-s + 0.197·35-s + 0.127·37-s − 0.394·38-s − 0.100·40-s − 1.59·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3922580025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3922580025\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.85T + 8T^{2} \) |
| 5 | \( 1 + 5.82T + 125T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 40.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 74.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 244.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 365.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 456.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 470.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 359.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 541.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−10.02807710294086412173407651179, −9.369553133563465427284524370280, −8.231248666633591103925035463511, −7.64402503980932719145798507113, −7.06753633136398318874858936043, −5.66008511873508182355819002144, −4.56287687440847775814121742433, −3.21427603445799429985494347293, −1.91627480023023785400789706772, −0.43129763167506888265700262038,
0.43129763167506888265700262038, 1.91627480023023785400789706772, 3.21427603445799429985494347293, 4.56287687440847775814121742433, 5.66008511873508182355819002144, 7.06753633136398318874858936043, 7.64402503980932719145798507113, 8.231248666633591103925035463511, 9.369553133563465427284524370280, 10.02807710294086412173407651179
