| L(s) = 1 | − 4·2-s + 25.4·3-s + 16·4-s + 17.9·5-s − 101.·6-s − 245.·7-s − 64·8-s + 403.·9-s − 71.7·10-s + 534.·11-s + 406.·12-s − 550.·13-s + 980.·14-s + 455.·15-s + 256·16-s + 611.·17-s − 1.61e3·18-s + 286.·20-s − 6.23e3·21-s − 2.13e3·22-s − 3.79e3·23-s − 1.62e3·24-s − 2.80e3·25-s + 2.20e3·26-s + 4.06e3·27-s − 3.92e3·28-s + 4.66e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s + 0.320·5-s − 1.15·6-s − 1.89·7-s − 0.353·8-s + 1.65·9-s − 0.226·10-s + 1.33·11-s + 0.815·12-s − 0.902·13-s + 1.33·14-s + 0.522·15-s + 0.250·16-s + 0.513·17-s − 1.17·18-s + 0.160·20-s − 3.08·21-s − 0.941·22-s − 1.49·23-s − 0.576·24-s − 0.897·25-s + 0.638·26-s + 1.07·27-s − 0.945·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 25.4T + 243T^{2} \) |
| 5 | \( 1 - 17.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 245.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 534.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 550.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 611.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.86e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 417.T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.55e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
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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
−9.407431965807927658132631669805, −8.505186222850154910307516841719, −7.68681068496864942053509389464, −6.70443861407326319941193989077, −6.14019422674942946395292166681, −4.17632517351290372194950128954, −3.27060569420058296167394268202, −2.59691006788503393052759911668, −1.48348071457409279363310176670, 0,
1.48348071457409279363310176670, 2.59691006788503393052759911668, 3.27060569420058296167394268202, 4.17632517351290372194950128954, 6.14019422674942946395292166681, 6.70443861407326319941193989077, 7.68681068496864942053509389464, 8.505186222850154910307516841719, 9.407431965807927658132631669805
