L(s) = 1 | − 2·4-s + 2.23i·5-s − 3.60·7-s − 2.23i·11-s + 3.60·13-s + 4·16-s − 8.06i·17-s − 4.47i·20-s − 8.06i·23-s − 5.00·25-s + 7.21·28-s − 8.06i·35-s − 3.60·37-s + 11.1i·41-s + 4.47i·44-s + ⋯ |
L(s) = 1 | − 4-s + 0.999i·5-s − 1.36·7-s − 0.674i·11-s + 1.00·13-s + 16-s − 1.95i·17-s − 0.999i·20-s − 1.68i·23-s − 1.00·25-s + 1.36·28-s − 1.36i·35-s − 0.592·37-s + 1.74i·41-s + 0.674i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.421540 - 0.421540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.421540 - 0.421540i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 17 | \( 1 + 8.06iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8.06iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 11.1iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.06iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 2.23iT - 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−10.35953442359925804597739457524, −9.598693601273356216215560229931, −8.913686922027174635693349457405, −7.87519005697604201429292484777, −6.66370143666822466027739805187, −6.12852378394196762989542180482, −4.82696543800890614048691086724, −3.52111941443955530834076750881, −2.91522951295656554951821953702, −0.36881477182371913175736337206,
1.43178131077830570229078081033, 3.58807539949549318511065618997, 4.12964500371094541039026739454, 5.48767134600582876150675813109, 6.11926811965491154687413270640, 7.51166875015286550108672545056, 8.567924488198617211728970963846, 9.117041957776312102367674665629, 9.874525492569226838286547095567, 10.64959093880742045827143833002