| L(s) = 1 | + (1.10 − 1.33i)3-s + (3.74 + 1.36i)5-s + (−0.452 + 2.56i)7-s + (−0.577 − 2.94i)9-s + (−4.99 + 1.81i)11-s + (−0.0404 + 0.0338i)13-s + (5.93 − 3.50i)15-s + (−1.69 − 2.92i)17-s + (1.23 − 2.13i)19-s + (2.93 + 3.43i)21-s + (−0.964 − 5.46i)23-s + (8.30 + 6.97i)25-s + (−4.57 − 2.46i)27-s + (−6.29 − 5.28i)29-s + (−0.115 − 0.656i)31-s + ⋯ |
| L(s) = 1 | + (0.635 − 0.772i)3-s + (1.67 + 0.608i)5-s + (−0.171 + 0.970i)7-s + (−0.192 − 0.981i)9-s + (−1.50 + 0.548i)11-s + (−0.0112 + 0.00940i)13-s + (1.53 − 0.904i)15-s + (−0.410 − 0.710i)17-s + (0.282 − 0.489i)19-s + (0.640 + 0.748i)21-s + (−0.201 − 1.14i)23-s + (1.66 + 1.39i)25-s + (−0.879 − 0.474i)27-s + (−1.16 − 0.981i)29-s + (−0.0207 − 0.117i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66669 - 0.170644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66669 - 0.170644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.10 + 1.33i)T \) |
| good | 5 | \( 1 + (-3.74 - 1.36i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.452 - 2.56i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.99 - 1.81i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0404 - 0.0338i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.69 + 2.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 2.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.964 + 5.46i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.29 + 5.28i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.115 + 0.656i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.67 - 4.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.31 - 4.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.0524 - 0.0190i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0794 - 0.450i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + (-3.99 - 1.45i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.457 + 2.59i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.16 + 6.01i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.18 - 12.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.15 - 12.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.93 - 4.97i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.83 - 1.54i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.04 + 3.53i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.8 - 4.32i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.68946874185804953404574218372, −11.36291690240025261884039303722, −10.03710304481079404716862107551, −9.454833797095819212104590776244, −8.364850872258142070723775692471, −7.11745079735991683801420708768, −6.17970416069813123109215601920, −5.22432949150952089811834980781, −2.67672369405781043851948942894, −2.26772519085734822532040097532,
2.00086587926171057960728817066, 3.54671138889949014278873433690, 5.07428048950087744448585030285, 5.79905177975136081288377511711, 7.49360278949667988378074864342, 8.614649441492371838372136641115, 9.537870033887101234578746939661, 10.31272512271763013010446723908, 10.85443073988361988212299919652, 12.83495899316473066182134604800
