L(s) = 1 | + (−1.79 + 2.40i)3-s − 4.55i·5-s + (5.85 − 10.1i)7-s + (−2.55 − 8.63i)9-s + (14.0 + 8.12i)11-s + (−3.63 + 6.29i)13-s + (10.9 + 8.17i)15-s + (−19.6 − 11.3i)17-s + (−13.5 − 13.2i)19-s + (13.8 + 32.2i)21-s + (−33.4 − 19.3i)23-s + 4.28·25-s + (25.3 + 9.37i)27-s + 25.8i·29-s + (14.0 + 24.3i)31-s + ⋯ |
L(s) = 1 | + (−0.598 + 0.801i)3-s − 0.910i·5-s + (0.835 − 1.44i)7-s + (−0.283 − 0.959i)9-s + (1.27 + 0.738i)11-s + (−0.279 + 0.483i)13-s + (0.729 + 0.544i)15-s + (−1.15 − 0.666i)17-s + (−0.715 − 0.698i)19-s + (0.659 + 1.53i)21-s + (−1.45 − 0.840i)23-s + 0.171·25-s + (0.937 + 0.347i)27-s + 0.890i·29-s + (0.452 + 0.784i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.015120456\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015120456\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.79 - 2.40i)T \) |
| 19 | \( 1 + (13.5 + 13.2i)T \) |
good | 5 | \( 1 + 4.55iT - 25T^{2} \) |
| 7 | \( 1 + (-5.85 + 10.1i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-14.0 - 8.12i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.63 - 6.29i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (19.6 + 11.3i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (33.4 + 19.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 25.8iT - 841T^{2} \) |
| 31 | \( 1 + (-14.0 - 24.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 47.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.35iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (39.1 + 67.8i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 16.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-43.6 + 25.1i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 - 74.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 62.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-28.5 + 49.4i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (73.0 + 42.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-28.4 + 49.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (59.5 + 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (71.4 + 41.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-123. + 71.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (59.2 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.21511619351761663319247013291, −9.027888312019707985696263636101, −8.651983069482349790035965952481, −7.07819346086720342596239946472, −6.62533459664506955656980235791, −4.98656215732478572574126024417, −4.51119891780744578604603650535, −3.91728381812094395178642772001, −1.71076779175834766134551103170, −0.38962953943912822250359540170,
1.67522423230247109388207556543, 2.53201080305898024838219689804, 4.06316349982794795741446902280, 5.49453294799601316217037148480, 6.14964857215750449416364873895, 6.77162975785056269014047572886, 8.163211639740591441414606201795, 8.434786722274866483325610805944, 9.772864238334421732807849624347, 10.85836501630578031039337388029