| L(s) = 1 | − 2.82·7-s + 3·9-s + 6.32i·11-s + 4.47i·13-s − 6.32i·19-s − 8.48·23-s + 4.47i·37-s − 2·41-s − 2.82·47-s + 1.00·49-s − 13.4i·53-s − 6.32i·59-s − 8.48·63-s − 17.8i·77-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.06·7-s + 9-s + 1.90i·11-s + 1.24i·13-s − 1.45i·19-s − 1.76·23-s + 0.735i·37-s − 0.312·41-s − 0.412·47-s + 0.142·49-s − 1.84i·53-s − 0.823i·59-s − 1.06·63-s − 2.03i·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4227328744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4227328744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 6.32iT - 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 6.32iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.334106563447187169811958415893, −8.278057911975440596036084505867, −7.27261599088309344996650071609, −6.82651069151560642388874071201, −6.36698517110010895693143815647, −4.98174706947258424915951031379, −4.40690917721682521284746137494, −3.69555219287335793129586724840, −2.38228239114615263251717266284, −1.67436883527775442014440289605,
0.12894277157611894695310244405, 1.33448767356358848593747205361, 2.77270984437400685218984501930, 3.55783863226895609980857454266, 4.10574934842512539013879318888, 5.61279634063926137773239755657, 5.89436892914822258699754208133, 6.65666998019224741444193455288, 7.76717751404731963636761878796, 8.126248881224206549591797507379
