| L(s) = 1 | + (1.55 + 2.09i)2-s + (−1.63 − 0.572i)3-s + (−1.37 + 4.59i)4-s + (−0.270 + 0.178i)5-s + (−1.34 − 4.31i)6-s + (2.64 − 2.80i)7-s + (−6.85 + 2.49i)8-s + (2.34 + 1.87i)9-s + (−0.794 − 0.289i)10-s + (1.12 − 0.565i)11-s + (4.88 − 6.72i)12-s + (−1.55 − 3.61i)13-s + (9.98 + 1.16i)14-s + (0.544 − 0.136i)15-s + (−7.87 − 5.18i)16-s + (−3.68 + 3.09i)17-s + ⋯ |
| L(s) = 1 | + (1.10 + 1.47i)2-s + (−0.943 − 0.330i)3-s + (−0.688 + 2.29i)4-s + (−0.121 + 0.0796i)5-s + (−0.550 − 1.75i)6-s + (0.999 − 1.05i)7-s + (−2.42 + 0.882i)8-s + (0.781 + 0.623i)9-s + (−0.251 − 0.0914i)10-s + (0.339 − 0.170i)11-s + (1.40 − 1.94i)12-s + (−0.432 − 1.00i)13-s + (2.66 + 0.311i)14-s + (0.140 − 0.0351i)15-s + (−1.96 − 1.29i)16-s + (−0.894 + 0.750i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.801313 + 0.927964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801313 + 0.927964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.63 + 0.572i)T \) |
| good | 2 | \( 1 + (-1.55 - 2.09i)T + (-0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (0.270 - 0.178i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-2.64 + 2.80i)T + (-0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 0.565i)T + (6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (1.55 + 3.61i)T + (-8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (3.68 - 3.09i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.47 + 2.91i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.385 - 0.408i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.11 + 0.247i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 0.876i)T + (27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (0.248 + 1.40i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (5.63 - 7.57i)T + (-11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.210 + 3.61i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-3.70 - 0.877i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (3.57 + 6.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.22 - 2.62i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 8.84i)T + (-50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (0.214 + 0.0250i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (11.2 + 4.07i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.37 + 2.68i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.98 - 6.69i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-0.396 - 0.531i)T + (-23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (-1.56 + 0.571i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.4 - 8.85i)T + (38.4 + 89.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.79801513675467149234641464032, −13.55922782299653674937410217588, −12.98170728687398709459793036691, −11.70391995094688313227518375682, −10.62283287198658891718786055532, −8.276237085663194310265531560349, −7.35559177677727032869374150120, −6.41766775713709532360634485052, −5.11778809616454163586768821341, −4.17474355401020573295843001920,
2.06542690432077585101605575920, 4.28763100674070514094255392205, 5.03408566557740097195217131327, 6.35353553758190530335487776804, 8.960233728451929133934411805963, 10.15889359292295289363194565316, 11.28559161706615654850595200267, 11.88140029814269186058577053588, 12.45954692415632925800498509221, 13.93395413430517153843726750611
