| L(s) = 1 | − 3.45·2-s + 3.95·4-s − 9.27·5-s + 27.2·7-s + 13.9·8-s + 32.0·10-s + 38.3·11-s − 94.1·14-s − 79.9·16-s + 85.0·17-s − 114.·19-s − 36.7·20-s − 132.·22-s + 49.4·23-s − 38.8·25-s + 107.·28-s − 11.5·29-s − 220.·31-s + 164.·32-s − 294.·34-s − 252.·35-s − 24.7·37-s + 395.·38-s − 129.·40-s − 30.8·41-s − 409.·43-s + 151.·44-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.494·4-s − 0.829·5-s + 1.47·7-s + 0.617·8-s + 1.01·10-s + 1.05·11-s − 1.79·14-s − 1.24·16-s + 1.21·17-s − 1.38·19-s − 0.410·20-s − 1.28·22-s + 0.448·23-s − 0.311·25-s + 0.727·28-s − 0.0740·29-s − 1.27·31-s + 0.910·32-s − 1.48·34-s − 1.22·35-s − 0.109·37-s + 1.68·38-s − 0.512·40-s − 0.117·41-s − 1.45·43-s + 0.520·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.45T + 8T^{2} \) |
| 5 | \( 1 + 9.27T + 125T^{2} \) |
| 7 | \( 1 - 27.2T + 343T^{2} \) |
| 11 | \( 1 - 38.3T + 1.33e3T^{2} \) |
| 17 | \( 1 - 85.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 24.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 409.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 434.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 716.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 213.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 238.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 748.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 258.T + 9.12e5T^{2} \) |
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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
−8.601737833324782717011208177564, −8.050333744957393082207464150231, −7.48839258705113855368691332255, −6.62383298036365800416368752246, −5.28833609585123975560216127527, −4.41843271839738749353643138451, −3.64854201012911612760753281563, −1.91420515897444940085359527251, −1.22374253243533967593818165086, 0,
1.22374253243533967593818165086, 1.91420515897444940085359527251, 3.64854201012911612760753281563, 4.41843271839738749353643138451, 5.28833609585123975560216127527, 6.62383298036365800416368752246, 7.48839258705113855368691332255, 8.050333744957393082207464150231, 8.601737833324782717011208177564
