L(s) = 1 | + (−1 + i)3-s + (1 − 2i)5-s + (3 + 3i)7-s + i·9-s − 2i·11-s + (3 + 3i)13-s + (1 + 3i)15-s + (1 − i)17-s − 4·19-s − 6·21-s + (1 − i)23-s + (−3 − 4i)25-s + (−4 − 4i)27-s − 10i·31-s + (2 + 2i)33-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.447 − 0.894i)5-s + (1.13 + 1.13i)7-s + 0.333i·9-s − 0.603i·11-s + (0.832 + 0.832i)13-s + (0.258 + 0.774i)15-s + (0.242 − 0.242i)17-s − 0.917·19-s − 1.30·21-s + (0.208 − 0.208i)23-s + (−0.600 − 0.800i)25-s + (−0.769 − 0.769i)27-s − 1.79i·31-s + (0.348 + 0.348i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07266 + 0.304722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07266 + 0.304722i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−12.95925770827375997058743013906, −11.63382711835295488583924754104, −11.28655009197773379600839819346, −9.945546337542022813772769662669, −8.772467860817053579338485519718, −8.206353204226717108679830505973, −6.12815307755934488180846606296, −5.28864812120296300610821443940, −4.38004981953623604220263144997, −1.95726066495511050852451073507,
1.51748122480919915396563750512, 3.65475271465668673728337717689, 5.28133571641083190207100039484, 6.58323582471142846291919349073, 7.28484508312509426651299023159, 8.488576150638774646669330552913, 10.22712027625269440624422582566, 10.74819187576907318581233985160, 11.69293023331710011895744125099, 12.84139341637347290761539981621