| L(s) = 1 | − 6.06·2-s − 9·3-s + 4.83·4-s + 26.0·5-s + 54.6·6-s − 152.·7-s + 164.·8-s + 81·9-s − 158.·10-s + 105.·11-s − 43.5·12-s − 582.·13-s + 926.·14-s − 234.·15-s − 1.15e3·16-s − 1.06e3·17-s − 491.·18-s + 390.·19-s + 126.·20-s + 1.37e3·21-s − 638.·22-s − 400.·23-s − 1.48e3·24-s − 2.44e3·25-s + 3.53e3·26-s − 729·27-s − 738.·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.151·4-s + 0.466·5-s + 0.619·6-s − 1.17·7-s + 0.910·8-s + 0.333·9-s − 0.500·10-s + 0.262·11-s − 0.0872·12-s − 0.955·13-s + 1.26·14-s − 0.269·15-s − 1.12·16-s − 0.897·17-s − 0.357·18-s + 0.248·19-s + 0.0704·20-s + 0.679·21-s − 0.281·22-s − 0.157·23-s − 0.525·24-s − 0.782·25-s + 1.02·26-s − 0.192·27-s − 0.177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4373878141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4373878141\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 6.06T + 32T^{2} \) |
| 5 | \( 1 - 26.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 152.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 105.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 582.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 390.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 400.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.72e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.79e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.61e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.26e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.26e5T + 8.58e9T^{2} \) |
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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
−11.55760648213782965858133606351, −10.51073894660982801985697957620, −9.587876989769454753560346129161, −9.221603599626544496392677119540, −7.66083410805255543645014575280, −6.73413676405898635390843607092, −5.55758886516735802781959422814, −4.08668470472481666428763318041, −2.15243772184414270060510925282, −0.49011152935370269740145836495,
0.49011152935370269740145836495, 2.15243772184414270060510925282, 4.08668470472481666428763318041, 5.55758886516735802781959422814, 6.73413676405898635390843607092, 7.66083410805255543645014575280, 9.221603599626544496392677119540, 9.587876989769454753560346129161, 10.51073894660982801985697957620, 11.55760648213782965858133606351
