| L(s) = 1 | − 9.91·2-s + 9·3-s + 66.3·4-s − 89.2·6-s − 92.6·7-s − 340.·8-s + 81·9-s + 121·11-s + 597.·12-s − 800.·13-s + 918.·14-s + 1.25e3·16-s + 117.·17-s − 803.·18-s + 831.·19-s − 833.·21-s − 1.20e3·22-s − 2.95e3·23-s − 3.06e3·24-s + 7.93e3·26-s + 729·27-s − 6.14e3·28-s + 5.76e3·29-s − 61.7·31-s − 1.56e3·32-s + 1.08e3·33-s − 1.16e3·34-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.07·4-s − 1.01·6-s − 0.714·7-s − 1.88·8-s + 0.333·9-s + 0.301·11-s + 1.19·12-s − 1.31·13-s + 1.25·14-s + 1.22·16-s + 0.0988·17-s − 0.584·18-s + 0.528·19-s − 0.412·21-s − 0.528·22-s − 1.16·23-s − 1.08·24-s + 2.30·26-s + 0.192·27-s − 1.48·28-s + 1.27·29-s − 0.0115·31-s − 0.270·32-s + 0.174·33-s − 0.173·34-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 | \( 1 + 9.91T + 32T^{2} \) |
| 7 | \( 1 + 92.6T + 1.68e4T^{2} \) |
| 13 | \( 1 + 800.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 117.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 831.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 61.7T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.59e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.74e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.48e4T + 8.58e9T^{2} \) |
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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
−9.211569740296842665116762806202, −8.257557615955186727819841351117, −7.60956329981179964046102051645, −6.88137841462313994356338786357, −5.98221147143337037534428017675, −4.40510189953053436500242680194, −2.98388976551263377331587014389, −2.25761977350130678574884441741, −1.04356352964306176142140735041, 0,
1.04356352964306176142140735041, 2.25761977350130678574884441741, 2.98388976551263377331587014389, 4.40510189953053436500242680194, 5.98221147143337037534428017675, 6.88137841462313994356338786357, 7.60956329981179964046102051645, 8.257557615955186727819841351117, 9.211569740296842665116762806202
