L(s) = 1 | + (1.89 + 1.09i)2-s + (0.895 − 1.55i)3-s + (1.39 + 2.41i)4-s − 2.18i·5-s + (3.39 − 1.96i)6-s + 1.73i·8-s + (−0.104 − 0.180i)9-s + (2.39 − 4.14i)10-s + (−1.10 − 0.637i)11-s + 4.99·12-s + (3.5 − 0.866i)13-s + (−3.39 − 1.96i)15-s + (0.895 − 1.55i)16-s + (−1.5 − 2.59i)17-s − 0.456i·18-s + (−5.68 + 3.28i)19-s + ⋯ |
L(s) = 1 | + (1.34 + 0.773i)2-s + (0.517 − 0.895i)3-s + (0.697 + 1.20i)4-s − 0.978i·5-s + (1.38 − 0.800i)6-s + 0.612i·8-s + (−0.0347 − 0.0602i)9-s + (0.757 − 1.31i)10-s + (−0.332 − 0.192i)11-s + 1.44·12-s + (0.970 − 0.240i)13-s + (−0.876 − 0.506i)15-s + (0.223 − 0.387i)16-s + (−0.363 − 0.630i)17-s − 0.107i·18-s + (−1.30 + 0.753i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.49286 - 0.471241i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.49286 - 0.471241i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.89 - 1.09i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 2.18iT - 5T^{2} \) |
| 11 | \( 1 + (1.10 + 0.637i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.68 - 3.28i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.79 - 6.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 1.91i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + (-6 - 3.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.20 - 1.27i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.28iT - 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + (7.66 - 4.42i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.87 - 5.70i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.791 - 0.456i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 3.55iT - 83T^{2} \) |
| 89 | \( 1 + (2.52 + 1.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 - 7.61i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.73920940764676020521578154728, −9.418201526979367910974328846522, −8.252230474317658347158637163226, −7.928027881827040283640195977636, −6.77265181825732844970368427372, −6.02767251626610196200092231380, −5.06120530117797538182440499266, −4.19278484603384417249220550813, −3.00009406879540505672048318217, −1.45806149572910146250777351636,
2.23740586056421171270311813393, 3.03231169592073939083722835950, 4.14259736532376176106522393614, 4.42662472598668251575783098717, 6.01455915238214103871027464167, 6.58536546964143114660212051529, 8.138070662658480822849275600281, 9.065220923886833842742234308146, 10.18997249602294660879466164360, 10.85459877679977747588240550140