| L(s) = 1 | + (−1.71 + 0.392i)2-s + (−0.804 + 0.387i)4-s + (−5.40 + 4.30i)5-s + 6.65i·7-s + (6.74 − 5.37i)8-s + (7.59 − 9.52i)10-s + (16.0 + 7.72i)11-s + (12.5 + 15.7i)13-s + (−2.60 − 11.4i)14-s + (−7.25 + 9.09i)16-s + (−12.5 + 15.7i)17-s + (−4.58 − 9.52i)19-s + (2.67 − 5.55i)20-s + (−30.5 − 6.97i)22-s + (−25.0 − 12.0i)23-s + ⋯ |
| L(s) = 1 | + (−0.859 + 0.196i)2-s + (−0.201 + 0.0968i)4-s + (−1.08 + 0.861i)5-s + 0.950i·7-s + (0.842 − 0.672i)8-s + (0.759 − 0.952i)10-s + (1.45 + 0.701i)11-s + (0.966 + 1.21i)13-s + (−0.186 − 0.816i)14-s + (−0.453 + 0.568i)16-s + (−0.737 + 0.925i)17-s + (−0.241 − 0.501i)19-s + (0.133 − 0.277i)20-s + (−1.38 − 0.317i)22-s + (−1.09 − 0.525i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0594402 - 0.516949i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0594402 - 0.516949i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (38.1 - 19.8i)T \) |
| good | 2 | \( 1 + (1.71 - 0.392i)T + (3.60 - 1.73i)T^{2} \) |
| 5 | \( 1 + (5.40 - 4.30i)T + (5.56 - 24.3i)T^{2} \) |
| 7 | \( 1 - 6.65iT - 49T^{2} \) |
| 11 | \( 1 + (-16.0 - 7.72i)T + (75.4 + 94.6i)T^{2} \) |
| 13 | \( 1 + (-12.5 - 15.7i)T + (-37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (12.5 - 15.7i)T + (-64.3 - 281. i)T^{2} \) |
| 19 | \( 1 + (4.58 + 9.52i)T + (-225. + 282. i)T^{2} \) |
| 23 | \( 1 + (25.0 + 12.0i)T + (329. + 413. i)T^{2} \) |
| 29 | \( 1 + (8.39 - 1.91i)T + (757. - 364. i)T^{2} \) |
| 31 | \( 1 + (-3.83 - 16.7i)T + (-865. + 416. i)T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-7.59 - 33.2i)T + (-1.51e3 + 729. i)T^{2} \) |
| 47 | \( 1 + (24.4 - 11.7i)T + (1.37e3 - 1.72e3i)T^{2} \) |
| 53 | \( 1 + (-54.8 + 68.8i)T + (-625. - 2.73e3i)T^{2} \) |
| 59 | \( 1 + (-19.1 + 23.9i)T + (-774. - 3.39e3i)T^{2} \) |
| 61 | \( 1 + (12.5 + 2.87i)T + (3.35e3 + 1.61e3i)T^{2} \) |
| 67 | \( 1 + (17.6 - 8.52i)T + (2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (22.2 + 46.3i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-6.25 + 4.98i)T + (1.18e3 - 5.19e3i)T^{2} \) |
| 79 | \( 1 + 8.15T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-4.73 + 20.7i)T + (-6.20e3 - 2.98e3i)T^{2} \) |
| 89 | \( 1 + (-62.1 - 14.1i)T + (7.13e3 + 3.43e3i)T^{2} \) |
| 97 | \( 1 + (53.1 + 25.5i)T + (5.86e3 + 7.35e3i)T^{2} \) |
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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
−11.53279033318377256596267877268, −10.69222624575739957607970009950, −9.486572415256705513665107017484, −8.802442996497082152030083816781, −8.127998544055416818374132035166, −6.87802118115616177648458050624, −6.43191491140368626486807627037, −4.34637951948447717268212316000, −3.75530564456315929278132578763, −1.82348879434481097563265852972,
0.37191428287510528497214340745, 1.22377291807646809355702133750, 3.71868246597227036931515747669, 4.35675631615153758018106935908, 5.75117334587453513489357104620, 7.20084781387883362381629164770, 8.198332333326496548360422476748, 8.624982192420871007584382885442, 9.608119154917991495748844090336, 10.60070129351406633757799021308
