| L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.00457 + 0.00545i)5-s + (2.74 + 2.29i)7-s + (−0.866 + 0.5i)8-s + (0.00355 − 0.00616i)10-s + (1.69 + 2.93i)11-s + (−0.226 − 0.620i)13-s + (−3.09 − 1.78i)14-s + (0.766 − 0.642i)16-s + (−0.200 + 0.552i)17-s + (−5.18 − 0.914i)19-s + (−0.00243 + 0.00668i)20-s + (−2.17 − 2.59i)22-s + (−1.53 − 0.885i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.469 − 0.171i)4-s + (−0.00204 + 0.00243i)5-s + (1.03 + 0.869i)7-s + (−0.306 + 0.176i)8-s + (0.00112 − 0.00194i)10-s + (0.510 + 0.884i)11-s + (−0.0626 − 0.172i)13-s + (−0.827 − 0.477i)14-s + (0.191 − 0.160i)16-s + (−0.0487 + 0.133i)17-s + (−1.19 − 0.209i)19-s + (−0.000544 + 0.00149i)20-s + (−0.464 − 0.553i)22-s + (−0.319 − 0.184i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968790 + 0.634522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968790 + 0.634522i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-2.25 - 5.64i)T \) |
| good | 5 | \( 1 + (0.00457 - 0.00545i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.74 - 2.29i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 2.93i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.226 + 0.620i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.200 - 0.552i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (5.18 + 0.914i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (1.53 + 0.885i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.13 + 4.11i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.88iT - 31T^{2} \) |
| 41 | \( 1 + (-0.731 + 0.266i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 + (4.47 - 7.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.17 + 3.50i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.64 - 9.10i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.544 - 1.49i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.11 - 5.12i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.24 - 7.04i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 4.67T + 73T^{2} \) |
| 79 | \( 1 + (-5.21 + 6.21i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.24 - 2.27i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.78 + 9.27i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (11.4 + 6.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.55979266459020140677012615434, −9.791887770932970109921265145915, −8.710058407262418790431200890276, −8.365421374536121759466645080852, −7.24121746250891981197119608090, −6.38304018341056790183507073899, −5.26609450460517697349549229097, −4.32441439496424417697100213936, −2.59001099697911593511124435216, −1.54398065032711641281998944970,
0.844037821325566350523204544570, 2.20783879719147515111177253238, 3.75972122299259410688001343689, 4.68903561742661488306608239013, 6.10192699593867113634192912832, 6.91703915629675989883801960192, 8.057567795683743100246857756955, 8.419708061541872746899910866784, 9.511157506877944072801937490456, 10.47847411040540274174953761373
