L(s) = 1 | + (0.340 − 1.69i)3-s + (1.68 + 0.296i)5-s + (−0.981 − 1.16i)7-s + (−2.76 − 1.15i)9-s + (−0.602 − 3.41i)11-s + (2.74 − 0.999i)13-s + (1.07 − 2.75i)15-s + (0.812 + 0.469i)17-s + (1.51 − 0.875i)19-s + (−2.32 + 1.26i)21-s + (0.0294 + 0.0247i)23-s + (−1.95 − 0.712i)25-s + (−2.90 + 4.30i)27-s + (−1.84 + 5.07i)29-s + (4.26 − 5.08i)31-s + ⋯ |
L(s) = 1 | + (0.196 − 0.980i)3-s + (0.752 + 0.132i)5-s + (−0.370 − 0.442i)7-s + (−0.922 − 0.385i)9-s + (−0.181 − 1.03i)11-s + (0.761 − 0.277i)13-s + (0.277 − 0.711i)15-s + (0.197 + 0.113i)17-s + (0.348 − 0.200i)19-s + (−0.506 + 0.276i)21-s + (0.00614 + 0.00515i)23-s + (−0.391 − 0.142i)25-s + (−0.559 + 0.828i)27-s + (−0.343 + 0.942i)29-s + (0.766 − 0.913i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07405 - 1.08678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07405 - 1.08678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.340 + 1.69i)T \) |
good | 5 | \( 1 + (-1.68 - 0.296i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.981 + 1.16i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.602 + 3.41i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.74 + 0.999i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.812 - 0.469i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.51 + 0.875i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0294 - 0.0247i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.84 - 5.07i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.26 + 5.08i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.09 + 5.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.60 - 4.41i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (8.03 - 1.41i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.29 + 5.27i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 12.6iT - 53T^{2} \) |
| 59 | \( 1 + (2.57 - 14.6i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 8.91i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.52 - 12.4i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (1.81 - 3.14i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.02 + 3.50i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.29 - 11.7i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.42 - 1.24i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.61 + 4.39i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.737 - 4.18i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.98992933537931609246356209430, −10.04197933587720385374309320152, −8.985518343316502180905927186764, −8.161603043595235570833808506932, −7.17154910347173566727300307323, −6.16790224496807567965676103028, −5.59699931135718728824954697218, −3.67910526610092153042000511153, −2.56604961321847992584571383434, −1.01207457729414354496858561296,
2.08559527929647455482875403895, 3.39417889074932330824459752106, 4.62645460754591494957417153871, 5.56989198703419491509766583011, 6.48711575523707369612578564900, 7.917636457851485282128444404368, 8.917860603491262067020834411135, 9.723882324122620809929116615855, 10.13703555977666565724098008644, 11.28908939961646953157269397320