| L(s) = 1 | + (0.468 − 1.33i)2-s + (−2.98 − 0.593i)3-s + (−1.56 − 1.24i)4-s + (0.914 − 1.36i)5-s + (−2.18 + 3.70i)6-s + (2.65 − 1.10i)7-s + (−2.39 + 1.49i)8-s + (5.76 + 2.38i)9-s + (−1.39 − 1.86i)10-s + (0.246 + 1.23i)11-s + (3.91 + 4.65i)12-s + (0.319 + 0.478i)13-s + (−0.224 − 4.05i)14-s + (−3.53 + 3.53i)15-s + (0.875 + 3.90i)16-s + (−3.38 − 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.331 − 0.943i)2-s + (−1.72 − 0.342i)3-s + (−0.780 − 0.624i)4-s + (0.408 − 0.611i)5-s + (−0.893 + 1.51i)6-s + (1.00 − 0.415i)7-s + (−0.848 + 0.529i)8-s + (1.92 + 0.796i)9-s + (−0.441 − 0.588i)10-s + (0.0742 + 0.373i)11-s + (1.12 + 1.34i)12-s + (0.0885 + 0.132i)13-s + (−0.0598 − 1.08i)14-s + (−0.913 + 0.913i)15-s + (0.218 + 0.975i)16-s + (−0.820 − 0.820i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.333250 - 0.597117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333250 - 0.597117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.468 + 1.33i)T \) |
| good | 3 | \( 1 + (2.98 + 0.593i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.914 + 1.36i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.65 + 1.10i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.246 - 1.23i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.319 - 0.478i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (3.38 + 3.38i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.19 + 2.80i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.178 + 0.429i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (1.02 - 5.14i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + (-0.447 - 0.299i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-2.44 + 5.89i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 0.757i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (5.99 + 5.99i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.810 - 4.07i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (1.03 - 1.55i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (6.47 + 1.28i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-6.01 - 1.19i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (4.20 - 1.74i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (0.911 + 0.377i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.152 + 0.152i)T - 79iT^{2} \) |
| 83 | \( 1 + (5.16 - 3.45i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-1.48 - 3.59i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−14.13655613105748462970965470962, −13.10780206726762208914975689572, −12.14729331188153665531778243985, −11.30817399885104850252991731216, −10.56212455801254011002943303607, −9.121957683353467440447689113232, −7.01722735273339452040088472285, −5.31151477989560884695400271994, −4.73710896371357978577284578847, −1.32790377460205662177969793514,
4.39907052815601588790798221687, 5.68204019056262347676063275423, 6.35023238820394430848880685444, 7.921336446008301764964088279002, 9.706628424221756690118220993201, 11.04344558822926493750803565883, 11.84910841212456964557371735504, 13.14242183866553038171321921695, 14.50465551492542259353929883140, 15.42438112460827635565549659813
