L(s) = 1 | + (0.258 + 0.965i)2-s + (1.95 + 0.523i)3-s + (−0.866 + 0.499i)4-s + (−1.82 − 1.29i)5-s + 2.02i·6-s + (−1.90 − 1.83i)7-s + (−0.707 − 0.707i)8-s + (0.941 + 0.543i)9-s + (0.774 − 2.09i)10-s + (2.01 + 3.49i)11-s + (−1.95 + 0.523i)12-s + (−0.204 + 0.204i)13-s + (1.28 − 2.31i)14-s + (−2.89 − 3.47i)15-s + (0.500 − 0.866i)16-s + (−0.527 + 1.97i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (1.12 + 0.302i)3-s + (−0.433 + 0.249i)4-s + (−0.816 − 0.577i)5-s + 0.825i·6-s + (−0.718 − 0.695i)7-s + (−0.249 − 0.249i)8-s + (0.313 + 0.181i)9-s + (0.244 − 0.663i)10-s + (0.609 + 1.05i)11-s + (−0.563 + 0.151i)12-s + (−0.0568 + 0.0568i)13-s + (0.343 − 0.618i)14-s + (−0.746 − 0.897i)15-s + (0.125 − 0.216i)16-s + (−0.128 + 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01775 + 0.439493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01775 + 0.439493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (1.82 + 1.29i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
good | 3 | \( 1 + (-1.95 - 0.523i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.204 - 0.204i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.527 - 1.97i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.10 + 5.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.38 - 1.17i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 7.15iT - 29T^{2} \) |
| 31 | \( 1 + (-6.33 + 3.65i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.19 - 4.46i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.58iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 + 4.97i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.304 - 0.0815i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.14 + 8.00i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.427 + 0.740i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.99 + 3.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 0.817i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + (11.1 + 2.98i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.39 - 2.53i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 + 3.85i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.53 + 2.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \) |
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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
−15.02114838066296149646306805153, −13.91890545644486055968246995273, −13.00986924943861798589532042553, −11.81441685812731266893856370687, −9.876278890333592223039078553262, −8.992457581677683318599991035069, −7.87393551760799619698812387548, −6.79736789480212432017013732695, −4.58764485028610811102852598859, −3.49343293478766282126562683169,
2.74262518317713777047060118223, 3.72151607898668227848276029120, 6.11789369285584648934890376812, 7.85323044676230143572932533137, 8.805808909188868843766211655383, 10.00212779268881526621358399677, 11.50745259284858768659497626760, 12.27325448783115957607170603931, 13.68948295113775847510920849330, 14.28096776120621046111758643679