| L(s) = 1 | + (−3.65 + 0.191i)2-s + (−3.56 + 4.40i)3-s + (5.37 − 0.565i)4-s + (−7.45 + 8.33i)5-s + (12.2 − 16.7i)6-s + (5.89 − 17.5i)7-s + (9.37 − 1.48i)8-s + (−1.07 − 5.04i)9-s + (25.6 − 31.9i)10-s + (45.0 + 9.57i)11-s + (−16.7 + 25.7i)12-s + (34.9 − 17.8i)13-s + (−18.1 + 65.3i)14-s + (−10.1 − 62.5i)15-s + (−76.3 + 16.2i)16-s + (19.9 − 51.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 + 0.0677i)2-s + (−0.686 + 0.848i)3-s + (0.672 − 0.0706i)4-s + (−0.666 + 0.745i)5-s + (0.830 − 1.14i)6-s + (0.318 − 0.948i)7-s + (0.414 − 0.0656i)8-s + (−0.0397 − 0.186i)9-s + (0.811 − 1.00i)10-s + (1.23 + 0.262i)11-s + (−0.401 + 0.618i)12-s + (0.746 − 0.380i)13-s + (−0.346 + 1.24i)14-s + (−0.174 − 1.07i)15-s + (−1.19 + 0.253i)16-s + (0.284 − 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.378484 + 0.456829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378484 + 0.456829i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (7.45 - 8.33i)T \) |
| 7 | \( 1 + (-5.89 + 17.5i)T \) |
| good | 2 | \( 1 + (3.65 - 0.191i)T + (7.95 - 0.836i)T^{2} \) |
| 3 | \( 1 + (3.56 - 4.40i)T + (-5.61 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-45.0 - 9.57i)T + (1.21e3 + 541. i)T^{2} \) |
| 13 | \( 1 + (-34.9 + 17.8i)T + (1.29e3 - 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-19.9 + 51.9i)T + (-3.65e3 - 3.28e3i)T^{2} \) |
| 19 | \( 1 + (3.07 - 29.2i)T + (-6.70e3 - 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-1.05 - 20.1i)T + (-1.21e4 + 1.27e3i)T^{2} \) |
| 29 | \( 1 + (-159. - 218. i)T + (-7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-50.0 - 112. i)T + (-1.99e4 + 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-12.2 - 7.98i)T + (2.06e4 + 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-8.43 - 2.74i)T + (5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + (140. - 140. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-28.8 + 11.0i)T + (7.71e4 - 6.94e4i)T^{2} \) |
| 53 | \( 1 + (-376. - 305. i)T + (3.09e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (181. + 201. i)T + (-2.14e4 + 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-204. - 184. i)T + (2.37e4 + 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-332. - 127. i)T + (2.23e5 + 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-562. + 408. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (473. + 729. i)T + (-1.58e5 + 3.55e5i)T^{2} \) |
| 79 | \( 1 + (500. - 1.12e3i)T + (-3.29e5 - 3.66e5i)T^{2} \) |
| 83 | \( 1 + (110. + 699. i)T + (-5.43e5 + 1.76e5i)T^{2} \) |
| 89 | \( 1 + (368. - 409. i)T + (-7.36e4 - 7.01e5i)T^{2} \) |
| 97 | \( 1 + (193. - 1.22e3i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−11.93848911306117597130812144048, −11.04611427077669010722181158146, −10.51986360993042396903117970042, −9.755250873058545779333170417421, −8.524620130728443113378565716023, −7.47286236245063476821324831228, −6.59867961872785414921219949651, −4.73671838827613333204018347146, −3.69940158688977874821531717367, −1.05622610383158196512504936737,
0.63336229715154040190483129029, 1.65058690429350924241670801204, 4.21898794463199620947518956340, 5.85871352352208008014374521800, 6.89917913859799221630545944404, 8.215649147852320810140586481138, 8.704980351279434287496856677042, 9.700699296583003574686203377875, 11.32644442786332927576842783226, 11.67345903677030602000470471197
