| L(s) = 1 | + (1.19 + 0.252i)2-s + (0.855 + 1.48i)3-s + (−0.474 − 0.211i)4-s + (0.320 + 3.05i)5-s + (0.643 + 1.97i)6-s + (−0.964 + 2.46i)7-s + (−2.48 − 1.80i)8-s + (0.0376 − 0.0652i)9-s + (−0.390 + 3.71i)10-s + (0.561 − 5.33i)11-s + (−0.0928 − 0.883i)12-s + (1.21 + 3.74i)13-s + (−1.77 + 2.68i)14-s + (−4.24 + 3.08i)15-s + (−1.80 − 2.00i)16-s + (−0.0522 + 0.497i)17-s + ⋯ |
| L(s) = 1 | + (0.841 + 0.178i)2-s + (0.493 + 0.855i)3-s + (−0.237 − 0.105i)4-s + (0.143 + 1.36i)5-s + (0.262 + 0.807i)6-s + (−0.364 + 0.931i)7-s + (−0.876 − 0.637i)8-s + (0.0125 − 0.0217i)9-s + (−0.123 + 1.17i)10-s + (0.169 − 1.60i)11-s + (−0.0268 − 0.254i)12-s + (0.337 + 1.03i)13-s + (−0.473 + 0.718i)14-s + (−1.09 + 0.795i)15-s + (−0.450 − 0.500i)16-s + (−0.0126 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0363 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0363 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35633 + 1.40663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35633 + 1.40663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.964 - 2.46i)T \) |
| 41 | \( 1 + (6.33 - 0.910i)T \) |
| good | 2 | \( 1 + (-1.19 - 0.252i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.855 - 1.48i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.320 - 3.05i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.561 + 5.33i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.21 - 3.74i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.0522 - 0.497i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.220 + 0.244i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-7.81 - 1.66i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-5.65 + 4.10i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.425 + 4.05i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.902 - 8.58i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (3.12 + 9.60i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (1.40 + 0.298i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (9.45 + 4.20i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (6.30 - 7.00i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (7.50 + 8.33i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.482 + 0.214i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-4.97 - 3.61i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.91 + 8.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.85 + 8.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.22T + 83T^{2} \) |
| 89 | \( 1 + (1.62 + 1.80i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-7.45 + 5.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.99183815106454195750472820419, −11.17431926049674909603635296508, −10.10200921769417621552920851794, −9.219840620059858153437409907267, −8.581497499924629213781959919016, −6.58504650843566067786262067584, −6.19329989369301357613470484516, −4.83062624826088639120956235887, −3.45809290520411935461776663231, −3.03277131842044418155546619629,
1.29008512951439457465515744741, 3.03470530890242545203361345889, 4.53417342465877779279818248143, 5.04565230252454106353756177456, 6.69480082353753108452302911411, 7.71864527467127630580954864467, 8.621622839253461251002927799763, 9.524892115920331945516680101501, 10.71563423357441673689870676626, 12.35615869028195788781813819884
