| L(s) = 1 | + 2.82i·5-s − i·7-s − 1.41·11-s − 6·13-s + 5.65i·17-s + 8i·19-s + 1.41·23-s − 3.00·25-s − 4.24i·29-s − 4i·31-s + 2.82·35-s − 8·37-s − 8.48i·41-s − 12i·43-s + 2.82·47-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s − 0.377i·7-s − 0.426·11-s − 1.66·13-s + 1.37i·17-s + 1.83i·19-s + 0.294·23-s − 0.600·25-s − 0.787i·29-s − 0.718i·31-s + 0.478·35-s − 1.31·37-s − 1.32i·41-s − 1.82i·43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.016495923756974396838573475046, −7.33505618527327242170781238094, −6.94124532454998661556275161589, −5.92962879205708946777875567888, −5.39883488010832727775566639967, −4.14571678259333694894240193715, −3.60493391855988244187056645824, −2.55863562662590887780279515853, −1.85662597664469685605532191669, 0,
1.10296259542620868119074977817, 2.44931022475125646472789536173, 3.02457742325209526002657234705, 4.56117198366236513425591792397, 5.04015736137516009135718947757, 5.25358561188571956025936718792, 6.69008292648942672707350513389, 7.20423037543759980742253573720, 8.031212910854721494207427498776
