| L(s) = 1 | + 8.73·2-s + 9·3-s + 44.2·4-s − 34.5·5-s + 78.5·6-s + 140.·7-s + 107.·8-s + 81·9-s − 302.·10-s + 57.6·11-s + 398.·12-s − 352.·13-s + 1.22e3·14-s − 311.·15-s − 481.·16-s − 1.58e3·17-s + 707.·18-s − 1.53e3·20-s + 1.26e3·21-s + 503.·22-s − 2.25e3·23-s + 963.·24-s − 1.92e3·25-s − 3.07e3·26-s + 729·27-s + 6.22e3·28-s − 1.71e3·29-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.38·4-s − 0.618·5-s + 0.891·6-s + 1.08·7-s + 0.591·8-s + 0.333·9-s − 0.955·10-s + 0.143·11-s + 0.798·12-s − 0.578·13-s + 1.67·14-s − 0.357·15-s − 0.469·16-s − 1.32·17-s + 0.514·18-s − 0.855·20-s + 0.626·21-s + 0.221·22-s − 0.889·23-s + 0.341·24-s − 0.617·25-s − 0.892·26-s + 0.192·27-s + 1.50·28-s − 0.378·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 8.73T + 32T^{2} \) |
| 5 | \( 1 + 34.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 140.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 57.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 352.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.28e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.21e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.23e3T + 8.58e9T^{2} \) |
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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
−8.575880306539819705298726730554, −7.72650039132923847655207265198, −7.02623648506932790127905212577, −5.96700091963856050016608607066, −5.00369585269591043171477730822, −4.27880778806936406504839403627, −3.73958523140937303457389658007, −2.51327375339796087185632981849, −1.80080843472178054560564813602, 0,
1.80080843472178054560564813602, 2.51327375339796087185632981849, 3.73958523140937303457389658007, 4.27880778806936406504839403627, 5.00369585269591043171477730822, 5.96700091963856050016608607066, 7.02623648506932790127905212577, 7.72650039132923847655207265198, 8.575880306539819705298726730554
