| L(s) = 1 | + (−1.36 − 1.94i)2-s + (1.63 − 0.583i)3-s + (−1.25 + 3.43i)4-s + (−3.36 − 2.38i)6-s + (2.19 + 1.02i)7-s + (3.81 − 1.02i)8-s + (2.32 − 1.90i)9-s + (0.00275 − 0.00327i)11-s + (−0.0362 + 6.34i)12-s + (1.13 + 0.792i)13-s + (−1.00 − 5.68i)14-s + (−1.59 − 1.33i)16-s + (−1.90 − 0.510i)17-s + (−6.87 − 1.92i)18-s + (6.69 + 3.86i)19-s + ⋯ |
| L(s) = 1 | + (−0.964 − 1.37i)2-s + (0.941 − 0.336i)3-s + (−0.626 + 1.71i)4-s + (−1.37 − 0.972i)6-s + (0.830 + 0.387i)7-s + (1.34 − 0.361i)8-s + (0.773 − 0.633i)9-s + (0.000829 − 0.000988i)11-s + (−0.0104 + 1.83i)12-s + (0.313 + 0.219i)13-s + (−0.267 − 1.51i)14-s + (−0.398 − 0.334i)16-s + (−0.461 − 0.123i)17-s + (−1.61 − 0.453i)18-s + (1.53 + 0.886i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.933235 - 1.03707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.933235 - 1.03707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 + 0.583i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (1.36 + 1.94i)T + (-0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-2.19 - 1.02i)T + (4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.00275 + 0.00327i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 0.792i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.90 + 0.510i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.69 - 3.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.12 - 6.70i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.74 + 9.87i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (5.62 + 2.04i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.451 + 1.68i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.95 - 0.520i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.18 + 0.628i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.871 + 1.86i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-1.25 - 1.25i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.763 - 0.640i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (6.39 - 2.32i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (5.21 - 7.45i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-8.32 + 4.80i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.744 + 2.77i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.95 + 1.22i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.34 - 5.84i)T + (28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (3.03 - 5.26i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.09 + 12.5i)T + (-95.5 + 16.8i)T^{2} \) |
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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
−10.05753707972975401070186674401, −9.425943208531429301354094200895, −8.793600272228715794112866312154, −7.927770732453441640476077185560, −7.38637757235446998977880549871, −5.70605722235965038367395553962, −4.14290134201411265888026504491, −3.20534470481559922014589902894, −2.13445994138232077638285501856, −1.25191980665928921888755830875,
1.26796561629944835127135526939, 3.06590982697925071678348207248, 4.60579609247682710338120445752, 5.33026865960802097632650017310, 6.79302016736963841139341165291, 7.35118335842436082736447219191, 8.164692539285586520257439460100, 8.870557387736146398709945551983, 9.374271950994848037807305331388, 10.49719678700455692790358965453
