| L(s) = 1 | + i·3-s + (−2 − i)5-s + i·7-s + 2·9-s + 3·11-s − i·13-s + (1 − 2i)15-s + 7i·17-s − 21-s + 6i·23-s + (3 + 4i)25-s + 5i·27-s + 5·29-s − 2·31-s + 3i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.377i·7-s + 0.666·9-s + 0.904·11-s − 0.277i·13-s + (0.258 − 0.516i)15-s + 1.69i·17-s − 0.218·21-s + 1.25i·23-s + (0.600 + 0.800i)25-s + 0.962i·27-s + 0.928·29-s − 0.359·31-s + 0.522i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11627 + 0.689896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11627 + 0.689896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.91070189481907922956927318156, −10.03091677087469056227918960742, −9.129171517005568276859253511243, −8.375848115054907090016583855751, −7.45311496359253204808358138568, −6.35389253844449468960839880985, −5.17906672504472111908101207858, −4.13719132333362772741355965576, −3.50828238529535548574726346732, −1.48207310990774027324936125367,
0.873193878614773183523832811103, 2.61115291801782959244242904270, 3.94513573682298272413934743563, 4.74636824709949268335118783326, 6.46838322567371333364017657592, 7.00666108758363224630411901529, 7.70808153712614068374976575634, 8.763191837037744281387501238879, 9.766656387781431151823788433392, 10.71009106781184143203315927908
