| L(s) = 1 | + (−0.173 − 0.984i)2-s + (1.70 + 0.284i)3-s + (−0.939 + 0.342i)4-s + (−0.550 − 0.461i)5-s + (−0.0164 − 1.73i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (2.83 + 0.972i)9-s + (−0.359 + 0.622i)10-s + (4.15 − 3.48i)11-s + (−1.70 + 0.316i)12-s + (0.378 − 2.14i)13-s + (−0.173 + 0.984i)14-s + (−0.808 − 0.945i)15-s + (0.766 − 0.642i)16-s + (1.25 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.986 + 0.164i)3-s + (−0.469 + 0.171i)4-s + (−0.246 − 0.206i)5-s + (−0.00670 − 0.707i)6-s + (−0.355 − 0.129i)7-s + (0.176 + 0.306i)8-s + (0.946 + 0.324i)9-s + (−0.113 + 0.196i)10-s + (1.25 − 1.05i)11-s + (−0.491 + 0.0914i)12-s + (0.105 − 0.596i)13-s + (−0.0464 + 0.263i)14-s + (−0.208 − 0.244i)15-s + (0.191 − 0.160i)16-s + (0.304 − 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40434 - 0.895297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40434 - 0.895297i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (-1.70 - 0.284i)T \) |
| 7 | \( 1 + (0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (0.550 + 0.461i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-4.15 + 3.48i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.378 + 2.14i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 2.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 1.84i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.34 + 0.855i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.39 - 7.91i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (8.08 - 2.94i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.93 - 6.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0296 + 0.167i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.34 - 3.64i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.35 - 1.95i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + (9.82 + 8.24i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (12.2 + 4.44i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.20 + 6.84i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.97 - 5.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.04 - 3.54i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.95 - 11.1i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.121 - 0.688i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.62 - 8.01i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.51 - 2.94i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.08285161649558345341284737251, −10.25439664505068827544400218338, −9.242214962828246163661560402010, −8.708963642540173325998735307985, −7.76431031383782220686768537350, −6.55614073635337424732686621826, −4.99952972899638198724300275599, −3.67174301274096394462447417805, −3.10347852341475193330179213754, −1.29423761762699650571797666740,
1.81147102043844429672400456390, 3.55618349284497265037179680665, 4.40459180931256429459270843697, 6.03268228125539059776427787635, 7.12855620851893049406935968812, 7.52561681972967086873083614021, 9.009037236044422671395251364004, 9.219904215962536923748299410913, 10.28207098905170861102603929542, 11.68417871411838916410009666099
