L(s) = 1 | + i·3-s + 1.58·5-s + (2.37 + 1.15i)7-s − 9-s + 2.26·11-s + 0.548·13-s + 1.58i·15-s − 0.433i·17-s − 6.02i·19-s + (−1.15 + 2.37i)21-s + 8.24i·23-s − 2.50·25-s − i·27-s + 0.548i·29-s + 7.50·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.706·5-s + (0.899 + 0.436i)7-s − 0.333·9-s + 0.681·11-s + 0.152·13-s + 0.408i·15-s − 0.105i·17-s − 1.38i·19-s + (−0.252 + 0.519i)21-s + 1.71i·23-s − 0.500·25-s − 0.192i·27-s + 0.101i·29-s + 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75306 + 0.716750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75306 + 0.716750i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-2.37 - 1.15i)T \) |
good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 0.548T + 13T^{2} \) |
| 17 | \( 1 + 0.433iT - 17T^{2} \) |
| 19 | \( 1 + 6.02iT - 19T^{2} \) |
| 23 | \( 1 - 8.24iT - 23T^{2} \) |
| 29 | \( 1 - 0.548iT - 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 - 4.21iT - 37T^{2} \) |
| 41 | \( 1 + 7.09iT - 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.71iT - 53T^{2} \) |
| 59 | \( 1 - 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 0.867iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6.13iT - 83T^{2} \) |
| 89 | \( 1 - 7.95iT - 89T^{2} \) |
| 97 | \( 1 + 19.1iT - 97T^{2} \) |
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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
−10.57763421441802753478298010364, −9.625480726535148070773801876173, −9.065548638709463017256888782327, −8.185443827083553392353087420453, −7.05155919211622804092057846629, −5.94134248969980748956293966424, −5.16409968496921792010113136909, −4.23304474297323862300249450602, −2.87134981596128243865511007512, −1.56141349966314318738156930747,
1.23761894764166714743548435675, 2.27388327698991915663565718862, 3.86120128894253535163784824185, 4.94743505288756497301622651562, 6.10646526743085691275496221421, 6.69730429512381388883631932680, 7.976783866518378680938970601442, 8.398420156953494987611266846532, 9.642339509640792908845054488566, 10.36022743259049973301372151614