| L(s) = 1 | + 0.746·7-s + 5.36i·11-s − 2.92i·13-s + 2.13·17-s − 1.73i·19-s − 7.49·23-s − 6.74i·29-s − 2.64·31-s + 1.07i·37-s + 11.2·41-s − 7.44i·43-s − 1.73·47-s − 6.44·49-s − 7.72i·53-s + 6.85i·59-s + ⋯ |
| L(s) = 1 | + 0.282·7-s + 1.61i·11-s − 0.811i·13-s + 0.517·17-s − 0.397i·19-s − 1.56·23-s − 1.25i·29-s − 0.475·31-s + 0.176i·37-s + 1.76·41-s − 1.13i·43-s − 0.252·47-s − 0.920·49-s − 1.06i·53-s + 0.892i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651409607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651409607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.746T + 7T^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 2.92iT - 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.07iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 7.72iT - 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 + 7.44iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85iT - 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.76653612157548767800604515614, −7.30053165425291285965124940544, −6.39493373277534485938987757883, −5.70853490782204386865836937244, −4.93613667615698850197661894804, −4.28257884901503494957299722220, −3.52039341283346133850690700034, −2.38708823682844956775393368226, −1.82222512835856594560231610175, −0.46036555188643315929775312818,
0.936023941472199371610259273760, 1.87591195794618289987691009591, 2.93603490346540873809114495365, 3.68696708679652720954638700257, 4.36454863262342347687067225346, 5.34951329294355020144828320117, 5.95741531534976521885018645184, 6.48143533202094751954955426146, 7.47798240767310601467200805548, 8.038755561124027375445514939024
