| L(s) = 1 | + (−2.83 − 1.03i)3-s + (−0.658 − 3.73i)5-s + (−0.0695 + 0.120i)7-s + (4.68 + 3.92i)9-s + (−0.350 − 0.607i)11-s + (−1.47 + 0.537i)13-s + (−1.98 + 11.2i)15-s + (2.88 − 2.41i)17-s + (4.28 − 0.790i)19-s + (0.321 − 0.269i)21-s + (1.02 − 5.82i)23-s + (−8.82 + 3.21i)25-s + (−4.69 − 8.13i)27-s + (5.28 + 4.43i)29-s + (1.43 − 2.49i)31-s + ⋯ |
| L(s) = 1 | + (−1.63 − 0.596i)3-s + (−0.294 − 1.67i)5-s + (−0.0262 + 0.0455i)7-s + (1.56 + 1.30i)9-s + (−0.105 − 0.183i)11-s + (−0.409 + 0.149i)13-s + (−0.513 + 2.91i)15-s + (0.698 − 0.586i)17-s + (0.983 − 0.181i)19-s + (0.0701 − 0.0588i)21-s + (0.214 − 1.21i)23-s + (−1.76 + 0.642i)25-s + (−0.903 − 1.56i)27-s + (0.980 + 0.823i)29-s + (0.258 − 0.447i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.299472 - 0.430989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.299472 - 0.430989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.28 + 0.790i)T \) |
| good | 3 | \( 1 + (2.83 + 1.03i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.658 + 3.73i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.0695 - 0.120i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.350 + 0.607i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.47 - 0.537i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.88 + 2.41i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.02 + 5.82i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.28 - 4.43i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.43 + 2.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.33T + 37T^{2} \) |
| 41 | \( 1 + (4.40 + 1.60i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.935 - 5.30i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.42 + 1.19i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.551 + 3.12i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 1.52i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.587 + 3.33i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.61 - 6.38i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.375 - 2.12i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 3.89i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.36 - 2.67i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.12 - 8.88i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (12.2 - 4.44i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.581 - 0.488i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.82540512571743764713674541751, −12.55459560063759348721561669917, −12.25593017139625495779945522906, −11.25122937268747657943433617557, −9.809975838614096279218766404515, −8.328261117514410151275645689079, −6.97063644647217323837100314086, −5.46101280272107241771523204431, −4.76278256281295231663374637781, −0.900443169310256046672168622395,
3.54873994653399418692293757922, 5.29614662047458069582827100335, 6.46713191209666503375188205377, 7.47958696268780775201769076632, 9.956043507099343276853386715295, 10.45326361769469748294660195247, 11.50492497784583396921503558258, 12.14721323779112419859493978284, 13.92882344810257654291744113187, 15.18233710699236755645467530603
