| L(s) = 1 | + 4.73·2-s + 14.3·4-s + 9.92·5-s + 7·7-s + 30.2·8-s + 46.9·10-s − 3.71·11-s − 15.5·13-s + 33.1·14-s + 28.0·16-s − 33.4·17-s + 135.·19-s + 142.·20-s − 17.5·22-s − 87.7·23-s − 26.4·25-s − 73.6·26-s + 100.·28-s + 242.·29-s − 194.·31-s − 109.·32-s − 158.·34-s + 69.4·35-s − 239.·37-s + 643.·38-s + 300.·40-s + 470.·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.79·4-s + 0.888·5-s + 0.377·7-s + 1.33·8-s + 1.48·10-s − 0.101·11-s − 0.332·13-s + 0.632·14-s + 0.437·16-s − 0.477·17-s + 1.64·19-s + 1.59·20-s − 0.170·22-s − 0.795·23-s − 0.211·25-s − 0.555·26-s + 0.679·28-s + 1.55·29-s − 1.12·31-s − 0.604·32-s − 0.799·34-s + 0.335·35-s − 1.06·37-s + 2.74·38-s + 1.18·40-s + 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.961205230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.961205230\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.73T + 8T^{2} \) |
| 5 | \( 1 - 9.92T + 125T^{2} \) |
| 11 | \( 1 + 3.71T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.15T + 1.03e5T^{2} \) |
| 53 | \( 1 + 736.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 279.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 44.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 901.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 487.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 726.T + 9.12e5T^{2} \) |
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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
−12.27632987659299655646781178799, −11.53596273464521670832837473914, −10.39527046502980608642461348633, −9.265470538002553362706371216565, −7.64659685835112797103145769696, −6.44846109697459173396428217355, −5.51343900559437254962157215947, −4.66444943384070729710683771265, −3.22541706504688024702033440526, −1.92515513340998566052728157358,
1.92515513340998566052728157358, 3.22541706504688024702033440526, 4.66444943384070729710683771265, 5.51343900559437254962157215947, 6.44846109697459173396428217355, 7.64659685835112797103145769696, 9.265470538002553362706371216565, 10.39527046502980608642461348633, 11.53596273464521670832837473914, 12.27632987659299655646781178799
