| L(s) = 1 | + (0.857 + 0.0449i)2-s + (−1.95 − 2.41i)3-s + (−1.25 − 0.131i)4-s + (−0.521 + 2.17i)5-s + (−1.56 − 2.15i)6-s + (−2.09 − 1.61i)7-s + (−2.76 − 0.438i)8-s + (−1.38 + 6.50i)9-s + (−0.544 + 1.84i)10-s + (2.41 − 0.513i)11-s + (2.13 + 3.29i)12-s + (−3.07 − 1.56i)13-s + (−1.72 − 1.47i)14-s + (6.26 − 2.99i)15-s + (0.117 + 0.0249i)16-s + (−1.30 − 3.40i)17-s + ⋯ |
| L(s) = 1 | + (0.606 + 0.0317i)2-s + (−1.12 − 1.39i)3-s + (−0.627 − 0.0659i)4-s + (−0.233 + 0.972i)5-s + (−0.640 − 0.880i)6-s + (−0.793 − 0.608i)7-s + (−0.978 − 0.154i)8-s + (−0.460 + 2.16i)9-s + (−0.172 + 0.582i)10-s + (0.728 − 0.154i)11-s + (0.616 + 0.949i)12-s + (−0.854 − 0.435i)13-s + (−0.461 − 0.394i)14-s + (1.61 − 0.772i)15-s + (0.0293 + 0.00624i)16-s + (−0.316 − 0.825i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00835839 - 0.360694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00835839 - 0.360694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.521 - 2.17i)T \) |
| 7 | \( 1 + (2.09 + 1.61i)T \) |
| good | 2 | \( 1 + (-0.857 - 0.0449i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (1.95 + 2.41i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-2.41 + 0.513i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (3.07 + 1.56i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.30 + 3.40i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.742 + 7.06i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.107 - 2.04i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.594 + 0.817i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.48 - 3.32i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.757 + 0.492i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-2.23 + 0.727i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.54 - 1.54i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.51 + 0.965i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (0.683 - 0.553i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (0.825 - 0.917i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (8.07 - 7.26i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.72 + 1.81i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (11.9 + 8.65i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.49 + 6.92i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (6.17 + 13.8i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.481 + 3.03i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (6.29 + 6.98i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-2.78 - 17.5i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.30176629823803375612245131208, −11.56447813369358873983280054848, −10.57048011215093599582952427779, −9.277927629555489449585438403458, −7.48493923686433028456727083717, −6.82330341598654033614592597071, −5.99968479950292101772761593265, −4.68346987071204700833172560918, −2.95102831098113879437635654741, −0.31025243709242205541084798253,
3.77899683311456966752959653427, 4.42364687106305354331073520052, 5.48912945944509543858765066423, 6.22746944501391492080689792432, 8.552885834456207763389615512429, 9.453094190283840455062574287507, 9.989060284236183874299541821012, 11.50793088314850862475931836297, 12.34874419398906326272886106015, 12.69172073578447049971300574925
