| L(s) = 1 | + (−2.08 − 0.109i)2-s + (−0.522 − 0.645i)3-s + (2.35 + 0.247i)4-s + (1.15 − 1.91i)5-s + (1.02 + 1.40i)6-s + (−1.53 + 2.15i)7-s + (−0.767 − 0.121i)8-s + (0.480 − 2.25i)9-s + (−2.61 + 3.87i)10-s + (1.02 − 0.218i)11-s + (−1.07 − 1.65i)12-s + (−4.08 − 2.08i)13-s + (3.43 − 4.33i)14-s + (−1.83 + 0.259i)15-s + (−3.05 − 0.648i)16-s + (−0.808 − 2.10i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 − 0.0773i)2-s + (−0.301 − 0.372i)3-s + (1.17 + 0.123i)4-s + (0.514 − 0.857i)5-s + (0.416 + 0.573i)6-s + (−0.578 + 0.815i)7-s + (−0.271 − 0.0429i)8-s + (0.160 − 0.753i)9-s + (−0.826 + 1.22i)10-s + (0.309 − 0.0658i)11-s + (−0.309 − 0.476i)12-s + (−1.13 − 0.576i)13-s + (0.917 − 1.15i)14-s + (−0.474 + 0.0669i)15-s + (−0.762 − 0.162i)16-s + (−0.196 − 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.207276 - 0.360298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.207276 - 0.360298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.15 + 1.91i)T \) |
| 7 | \( 1 + (1.53 - 2.15i)T \) |
| good | 2 | \( 1 + (2.08 + 0.109i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.522 + 0.645i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.02 + 0.218i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (4.08 + 2.08i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.808 + 2.10i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.520 + 4.95i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.325 + 6.21i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (0.743 - 1.02i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.14 - 9.30i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-5.27 + 3.42i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-5.32 + 1.72i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.80 - 4.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.292 - 0.112i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (0.217 - 0.176i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-4.60 + 5.10i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (3.94 - 3.54i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 4.40i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (1.66 + 1.21i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.705 - 1.08i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-4.40 - 9.89i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.579 + 3.66i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (7.09 + 7.88i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.775 - 4.89i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.44726637983619033377961124412, −11.26804222401800146732788054320, −9.995899769024630352779612250690, −9.219726101741418026462953246107, −8.747384081224042864929566871157, −7.31373198820898349899991275549, −6.32865192128933979834614412303, −4.93845354345741177491605910609, −2.44336130855628570487791401299, −0.62528622972363830762816323416,
2.04077445859095896719620692889, 4.10657378085350231198272163881, 5.96133894689628738653610492216, 7.19519902023888544789334216266, 7.76952956264058566377823265653, 9.531803884530938415456409895221, 9.841424577074642953057877984580, 10.70319588278993512729038435537, 11.46345889475111038229130662719, 13.11399039740614589734375981634
