| L(s) = 1 | + 0.936i·7-s − 2.20i·11-s + 3.33i·13-s − 1.54i·17-s − 3.12·19-s − 3.39·23-s + 8.44·29-s + 8.30i·31-s − 7.60i·37-s + 5.83i·41-s − 7.77·43-s + 10.7·47-s + 6.12·49-s − 5.08·53-s − 10.6i·59-s + ⋯ |
| L(s) = 1 | + 0.353i·7-s − 0.665i·11-s + 0.924i·13-s − 0.374i·17-s − 0.716·19-s − 0.707·23-s + 1.56·29-s + 1.49i·31-s − 1.25i·37-s + 0.910i·41-s − 1.18·43-s + 1.56·47-s + 0.874·49-s − 0.698·53-s − 1.39i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8877723729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8877723729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.936iT - 7T^{2} \) |
| 11 | \( 1 + 2.20iT - 11T^{2} \) |
| 13 | \( 1 - 3.33iT - 13T^{2} \) |
| 17 | \( 1 + 1.54iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 8.30iT - 31T^{2} \) |
| 37 | \( 1 + 7.60iT - 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 5.59T + 73T^{2} \) |
| 79 | \( 1 + 1.02iT - 79T^{2} \) |
| 83 | \( 1 - 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−8.303144895657518151234315307724, −7.47745615309283557552371798930, −6.63697035365365930047603166082, −6.23229918897546497723500255582, −5.35027219036483606173929446782, −4.62294921688031720351532711926, −3.88756683292936401859155189426, −2.97378664979804242014955758286, −2.19559280908990185372556119154, −1.15857348082855777483251748741,
0.22156941940987894883114763038, 1.42494464979043150464659593326, 2.42063925597305950682218595239, 3.21143380902799181274179992764, 4.27472974795485769363298606627, 4.58106015345122436386659158658, 5.75410135631135581808368587960, 6.13922736947270694510296590997, 7.11495309195383470387061312311, 7.59472433124580713613987003878
