| L(s) = 1 | + (−0.571 − 1.28i)2-s + (−0.0425 − 0.200i)3-s + (0.0186 − 0.0207i)4-s + (1.51 − 0.158i)5-s + (−0.232 + 0.168i)6-s + (−2.52 − 0.794i)7-s + (−2.70 − 0.880i)8-s + (2.70 − 1.20i)9-s + (−1.06 − 1.84i)10-s + (−2.37 + 2.31i)11-s + (−0.00494 − 0.00285i)12-s + (5.09 + 3.70i)13-s + (0.422 + 3.69i)14-s + (−0.0960 − 0.295i)15-s + (0.412 + 3.92i)16-s + (−2.50 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.403 − 0.907i)2-s + (−0.0245 − 0.115i)3-s + (0.00934 − 0.0103i)4-s + (0.676 − 0.0710i)5-s + (−0.0948 + 0.0689i)6-s + (−0.953 − 0.300i)7-s + (−0.957 − 0.311i)8-s + (0.900 − 0.401i)9-s + (−0.337 − 0.584i)10-s + (−0.716 + 0.697i)11-s + (−0.00142 − 0.000824i)12-s + (1.41 + 1.02i)13-s + (0.112 + 0.986i)14-s + (−0.0248 − 0.0763i)15-s + (0.103 + 0.980i)16-s + (−0.607 − 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0611 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0611 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.612554 - 0.576199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612554 - 0.576199i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.52 + 0.794i)T \) |
| 11 | \( 1 + (2.37 - 2.31i)T \) |
| good | 2 | \( 1 + (0.571 + 1.28i)T + (-1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.0425 + 0.200i)T + (-2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (-1.51 + 0.158i)T + (4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-5.09 - 3.70i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.50 + 1.11i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.41 - 4.90i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.513 + 0.888i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.40 - 1.10i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.552 - 0.0580i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (4.25 + 0.904i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.25 + 6.93i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.51iT - 43T^{2} \) |
| 47 | \( 1 + (5.45 - 4.91i)T + (4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.314 - 2.98i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (0.536 + 0.483i)T + (6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.822 + 7.82i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (2.16 + 3.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.87 - 2.81i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (6.52 - 7.24i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (1.12 + 2.52i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-5.85 + 4.25i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.985 - 0.569i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.71 - 6.49i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−13.88342159071196590659389409199, −12.97360203281395171892869458813, −12.04005916685514961028639839463, −10.70099293009594600078735282079, −9.843955151263808426642014382344, −9.140479759377309046992172505860, −7.06022183295384494248137627888, −5.96872662312263481241457082308, −3.70420027854498194499332860477, −1.75647227238261947359926734658,
3.08771858120556352862480163442, 5.56775732946422863456726417206, 6.47804557852478876571044345460, 7.79034772353414404221821592468, 8.964171337061114999792309013675, 10.07457463231123719435544553730, 11.30757582503776884564184378788, 13.08836721011903795926557200252, 13.41294063461060230931740445865, 15.30273898116630005540886871586
