| L(s) = 1 | + (0.219 − 4.89i)2-s + (−3.24 − 2.83i)3-s + (−15.9 − 1.43i)4-s + (6.12 + 18.8i)5-s + (−14.5 + 15.2i)6-s + (−30.0 − 8.30i)7-s + (−5.23 + 38.6i)8-s + (−1.12 − 8.33i)9-s + (93.5 − 25.8i)10-s + (−7.22 + 13.4i)11-s + (47.5 + 49.7i)12-s + (−19.3 − 35.9i)13-s + (−47.2 + 145. i)14-s + (33.6 − 78.6i)15-s + (62.2 + 11.2i)16-s + (27.7 + 20.1i)17-s + ⋯ |
| L(s) = 1 | + (0.0776 − 1.72i)2-s + (−0.624 − 0.545i)3-s + (−1.98 − 0.178i)4-s + (0.548 + 1.68i)5-s + (−0.992 + 1.03i)6-s + (−1.62 − 0.448i)7-s + (−0.231 + 1.70i)8-s + (−0.0418 − 0.308i)9-s + (2.95 − 0.816i)10-s + (−0.198 + 0.368i)11-s + (1.14 + 1.19i)12-s + (−0.412 − 0.767i)13-s + (−0.901 + 2.77i)14-s + (0.578 − 1.35i)15-s + (0.972 + 0.176i)16-s + (0.395 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0506 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0506 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.203433 + 0.214018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203433 + 0.214018i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-549. - 237. i)T \) |
| good | 2 | \( 1 + (-0.219 + 4.89i)T + (-7.96 - 0.717i)T^{2} \) |
| 3 | \( 1 + (3.24 + 2.83i)T + (3.62 + 26.7i)T^{2} \) |
| 5 | \( 1 + (-6.12 - 18.8i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (30.0 + 8.30i)T + (294. + 175. i)T^{2} \) |
| 11 | \( 1 + (7.22 - 13.4i)T + (-733. - 1.11e3i)T^{2} \) |
| 13 | \( 1 + (19.3 + 35.9i)T + (-1.21e3 + 1.83e3i)T^{2} \) |
| 17 | \( 1 + (-27.7 - 20.1i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (18.7 + 43.9i)T + (-4.73e3 + 4.95e3i)T^{2} \) |
| 23 | \( 1 + (77.8 + 97.5i)T + (-2.70e3 + 1.18e4i)T^{2} \) |
| 29 | \( 1 + (30.3 - 18.1i)T + (1.15e4 - 2.14e4i)T^{2} \) |
| 31 | \( 1 + (-55.2 + 10.0i)T + (2.78e4 - 1.04e4i)T^{2} \) |
| 37 | \( 1 + (-174. + 218. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 + (-239. - 115. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (495. + 185. i)T + (5.98e4 + 5.23e4i)T^{2} \) |
| 47 | \( 1 + (-403. + 352. i)T + (1.39e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (181. - 16.3i)T + (1.46e5 - 2.65e4i)T^{2} \) |
| 59 | \( 1 + (329. + 345. i)T + (-9.21e3 + 2.05e5i)T^{2} \) |
| 61 | \( 1 + (614. - 169. i)T + (1.94e5 - 1.16e5i)T^{2} \) |
| 67 | \( 1 + (-140. - 12.6i)T + (2.95e5 + 5.37e4i)T^{2} \) |
| 73 | \( 1 + (-22.3 + 496. i)T + (-3.87e5 - 3.48e4i)T^{2} \) |
| 79 | \( 1 + (98.4 - 727. i)T + (-4.75e5 - 1.31e5i)T^{2} \) |
| 83 | \( 1 + (-49.1 - 51.4i)T + (-2.56e4 + 5.71e5i)T^{2} \) |
| 89 | \( 1 + (1.34e3 - 121. i)T + (6.93e5 - 1.25e5i)T^{2} \) |
| 97 | \( 1 + (-1.02e3 + 491. i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−12.94453280937937739999191040774, −12.29527897878619489121562964529, −10.99601405625138520445167172400, −10.23559662135146965387021652649, −9.610368065449738299099267230689, −7.06005119751327175386143930213, −6.06350966326795177994334376106, −3.55832361973515894430648809219, −2.54771569573385443028714665330, −0.19382260340952079172724089938,
4.46295960534101377519893659446, 5.55684693928792081377966217059, 6.20114906227941333954141871760, 7.983290070608195926230801688936, 9.241928944146446232735188736937, 9.778715184204997091753825386181, 12.09927129791109290649862970601, 13.15668558377305089889331409700, 13.87644919778558542087591129357, 15.58608188279926193073773606315
