| L(s) = 1 | + (−1.39 + 0.245i)2-s + (1.87 − 0.684i)4-s + (3.55 − 4.24i)5-s + (−10.2 − 3.73i)7-s + (−2.44 + 1.41i)8-s + (−3.91 + 6.77i)10-s + (−2.64 − 3.15i)11-s + (−0.953 + 5.40i)13-s + (15.2 + 2.68i)14-s + (3.06 − 2.57i)16-s + (−4.10 − 2.37i)17-s + (−17.1 − 29.7i)19-s + (3.78 − 10.4i)20-s + (4.45 + 3.73i)22-s + (−12.6 − 34.6i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.469 − 0.171i)4-s + (0.711 − 0.848i)5-s + (−1.46 − 0.533i)7-s + (−0.306 + 0.176i)8-s + (−0.391 + 0.677i)10-s + (−0.240 − 0.286i)11-s + (−0.0733 + 0.416i)13-s + (1.08 + 0.191i)14-s + (0.191 − 0.160i)16-s + (−0.241 − 0.139i)17-s + (−0.903 − 1.56i)19-s + (0.189 − 0.520i)20-s + (0.202 + 0.169i)22-s + (−0.549 − 1.50i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.370469 - 0.597451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.370469 - 0.597451i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.55 + 4.24i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (10.2 + 3.73i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (2.64 + 3.15i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.953 - 5.40i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (4.10 + 2.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (17.1 + 29.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (12.6 + 34.6i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-5.18 + 0.914i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-34.9 + 12.7i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (12.1 - 21.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-22.6 - 3.98i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-39.1 + 32.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (28.3 - 77.8i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 16.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-45.6 + 54.4i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (74.1 + 26.9i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (12.1 - 68.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-65.9 - 38.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-25.5 - 44.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-5.51 - 31.2i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (28.7 - 5.06i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-69.2 + 40.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-36.3 + 30.5i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−12.52747458537102333701892370478, −11.04796564925020513060453298594, −10.02458925860247147390918321520, −9.295827807160999030397052988285, −8.444623675263957311771464073506, −6.86644309687069671455767553528, −6.15034017929166732362867991246, −4.53482168597553429920439252818, −2.60003400387890340820153765951, −0.51459846245262011625202253476,
2.24239012929963663707568237896, 3.45507839456800517731368790966, 5.87174627116221430332415235574, 6.50235233514302972331456159410, 7.78454082694629316956801279525, 9.124735609278658853354472008016, 10.06058691992593162926386036653, 10.46782997029703590979589756714, 11.98544551760980393091778392326, 12.80765772735705289364220633729
