| L(s) = 1 | + 7.09·2-s + 9·3-s + 18.3·4-s + 63.8·6-s − 4.62·7-s − 97.0·8-s + 81·9-s − 121·11-s + 164.·12-s + 38.0·13-s − 32.8·14-s − 1.27e3·16-s + 610.·17-s + 574.·18-s + 2.10e3·19-s − 41.6·21-s − 858.·22-s − 3.87e3·23-s − 873.·24-s + 270.·26-s + 729·27-s − 84.7·28-s − 8.37e3·29-s − 1.41e3·31-s − 5.93e3·32-s − 1.08e3·33-s + 4.32e3·34-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.572·4-s + 0.724·6-s − 0.0356·7-s − 0.536·8-s + 0.333·9-s − 0.301·11-s + 0.330·12-s + 0.0625·13-s − 0.0447·14-s − 1.24·16-s + 0.512·17-s + 0.418·18-s + 1.33·19-s − 0.0205·21-s − 0.378·22-s − 1.52·23-s − 0.309·24-s + 0.0784·26-s + 0.192·27-s − 0.0204·28-s − 1.84·29-s − 0.264·31-s − 1.02·32-s − 0.174·33-s + 0.642·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 7.09T + 32T^{2} \) |
| 7 | \( 1 + 4.62T + 1.68e4T^{2} \) |
| 13 | \( 1 - 38.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 610.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 591.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.59e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.35e4T + 8.58e9T^{2} \) |
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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
−9.219087777896102284022821175255, −7.995071410393264202863963530019, −7.35337996586138161589641893193, −6.08333496986454895429791704863, −5.47401663427822737102601295207, −4.43076309499044972429112926604, −3.59136877154359754826963741551, −2.84753625680686007572153734585, −1.65804331352346114337219006402, 0,
1.65804331352346114337219006402, 2.84753625680686007572153734585, 3.59136877154359754826963741551, 4.43076309499044972429112926604, 5.47401663427822737102601295207, 6.08333496986454895429791704863, 7.35337996586138161589641893193, 7.995071410393264202863963530019, 9.219087777896102284022821175255
