| L(s) = 1 | − 2.53·2-s − 1.55·4-s − 8.18·5-s + 10.7·7-s + 24.2·8-s + 20.7·10-s − 10.9·11-s − 27.3·14-s − 49.1·16-s − 4.85·17-s − 65.2·19-s + 12.7·20-s + 27.9·22-s + 166.·23-s − 57.9·25-s − 16.7·28-s − 92.4·29-s + 3.63·31-s − 69.3·32-s + 12.3·34-s − 88.3·35-s − 40.6·37-s + 165.·38-s − 198.·40-s + 242.·41-s + 161.·43-s + 17.0·44-s + ⋯ |
| L(s) = 1 | − 0.897·2-s − 0.194·4-s − 0.732·5-s + 0.582·7-s + 1.07·8-s + 0.657·10-s − 0.301·11-s − 0.522·14-s − 0.767·16-s − 0.0692·17-s − 0.788·19-s + 0.142·20-s + 0.270·22-s + 1.51·23-s − 0.463·25-s − 0.113·28-s − 0.592·29-s + 0.0210·31-s − 0.383·32-s + 0.0621·34-s − 0.426·35-s − 0.180·37-s + 0.707·38-s − 0.785·40-s + 0.925·41-s + 0.572·43-s + 0.0585·44-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 | \( 1 + 2.53T + 8T^{2} \) |
| 5 | \( 1 + 8.18T + 125T^{2} \) |
| 7 | \( 1 - 10.7T + 343T^{2} \) |
| 11 | \( 1 + 10.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.85T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 92.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.63T + 2.97e4T^{2} \) |
| 37 | \( 1 + 40.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 161.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 662.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 391.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 79.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 480.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−8.788677827550818223259126031553, −7.81539532051388888873149072958, −7.60288680879099963281741535649, −6.47948876193873221293313776629, −5.20494820055504738576383361751, −4.51203159338169483871332061595, −3.61780835628500726659386492881, −2.21622783226654669147075801708, −1.03484266777324195215219291139, 0,
1.03484266777324195215219291139, 2.21622783226654669147075801708, 3.61780835628500726659386492881, 4.51203159338169483871332061595, 5.20494820055504738576383361751, 6.47948876193873221293313776629, 7.60288680879099963281741535649, 7.81539532051388888873149072958, 8.788677827550818223259126031553
