| L(s) = 1 | + (1 + i)3-s + (−2 + i)5-s + (−1 + i)7-s − i·9-s − 4i·11-s + (3 − 3i)13-s + (−3 − i)15-s + (−3 − 3i)17-s + 6·19-s − 2·21-s + (−3 − 3i)23-s + (3 − 4i)25-s + (4 − 4i)27-s + 2i·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.577i)3-s + (−0.894 + 0.447i)5-s + (−0.377 + 0.377i)7-s − 0.333i·9-s − 1.20i·11-s + (0.832 − 0.832i)13-s + (−0.774 − 0.258i)15-s + (−0.727 − 0.727i)17-s + 1.37·19-s − 0.436·21-s + (−0.625 − 0.625i)23-s + (0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s + 0.371i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500711276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500711276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 23iT^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (-9 + 9i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + (-9 + 9i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (3 + 3i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (7 + 7i)T + 97iT^{2} \) |
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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
−9.520284011433220289967643089439, −8.617914218246992681510748451867, −8.319963941414979635705062489178, −7.16343907583786005929418723647, −6.32522873187797143820731578818, −5.38203084575206009875689790453, −4.16015594184457010061608941458, −3.25006326354692888511777464048, −2.93550550580576916165475114295, −0.65582297781718012055007100399,
1.30235978816778227887700498647, 2.39747503569600525383894734055, 3.81078526827458449596811091803, 4.31392515244849746139924828302, 5.53429960314262907686507309015, 6.76078023017890266240201548927, 7.43613424880558753341772018411, 7.954351199074866528575922904573, 8.873495929082855836628155208453, 9.539994810994042521246872505391
