| L(s) = 1 | − 3.60·2-s − 3·3-s + 8.99·4-s + 7.21·5-s + 10.8·6-s − 18.0·8-s + 9·9-s − 25.9·10-s + 7.21·11-s − 26.9·12-s + 13·13-s − 21.6·15-s + 28.9·16-s − 32.4·18-s + 64.8·20-s − 25.9·22-s + 54.0·24-s + 26.9·25-s − 46.8·26-s − 27·27-s + 77.9·30-s − 32.4·32-s − 21.6·33-s + 80.9·36-s − 39·39-s − 129.·40-s − 79.3·41-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 3-s + 2.24·4-s + 1.44·5-s + 1.80·6-s − 2.25·8-s + 9-s − 2.59·10-s + 0.655·11-s − 2.24·12-s + 13-s − 1.44·15-s + 1.81·16-s − 1.80·18-s + 3.24·20-s − 1.18·22-s + 2.25·24-s + 1.07·25-s − 1.80·26-s − 27-s + 2.59·30-s − 1.01·32-s − 0.655·33-s + 2.24·36-s − 39-s − 3.24·40-s − 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4863669376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4863669376\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 3.60T + 4T^{2} \) |
| 5 | \( 1 - 7.21T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 7.21T + 121T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 79.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70T + 1.84e3T^{2} \) |
| 47 | \( 1 - 93.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.21T + 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 79.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 50T + 6.24e3T^{2} \) |
| 83 | \( 1 + 165.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 79.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−16.75050293489166986658064912287, −15.47406542316161050872864157807, −13.54174296910171702203636830928, −11.87725115993539025017921452715, −10.68974787484638529313988094484, −9.885315306219304593499320891502, −8.784336294135615895615079488499, −6.88870809759334548929486050282, −5.89181887453980673580818070998, −1.51443530488159822255553156382,
1.51443530488159822255553156382, 5.89181887453980673580818070998, 6.88870809759334548929486050282, 8.784336294135615895615079488499, 9.885315306219304593499320891502, 10.68974787484638529313988094484, 11.87725115993539025017921452715, 13.54174296910171702203636830928, 15.47406542316161050872864157807, 16.75050293489166986658064912287
