L(s) = 1 | + (0.456 + 0.456i)3-s + (2.18 + 0.456i)5-s + (2.79 + 2.79i)7-s − 2.58i·9-s − 4.37·11-s + (−1.73 + 1.73i)13-s + (0.791 + 1.20i)15-s + (3 − 3i)17-s + 3.46i·19-s + 2.55i·21-s + (0.791 − 0.791i)23-s + (4.58 + 1.99i)25-s + (2.55 − 2.55i)27-s + 5.29·29-s + 1.58i·31-s + ⋯ |
L(s) = 1 | + (0.263 + 0.263i)3-s + (0.978 + 0.204i)5-s + (1.05 + 1.05i)7-s − 0.860i·9-s − 1.31·11-s + (−0.480 + 0.480i)13-s + (0.204 + 0.312i)15-s + (0.727 − 0.727i)17-s + 0.794i·19-s + 0.556i·21-s + (0.164 − 0.164i)23-s + (0.916 + 0.399i)25-s + (0.490 − 0.490i)27-s + 0.982·29-s + 0.284i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63729 + 0.461662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63729 + 0.461662i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.18 - 0.456i)T \) |
good | 3 | \( 1 + (-0.456 - 0.456i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + (1.73 - 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (-0.791 + 0.791i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 1.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 + 5.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + (8.29 + 8.29i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.791 + 0.791i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.64 - 2.64i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 9.66iT - 61T^{2} \) |
| 67 | \( 1 + (-8.29 + 8.29i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-0.582 - 0.582i)T + 73iT^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + (-4.83 - 4.83i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.16iT - 89T^{2} \) |
| 97 | \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \) |
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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
−11.89747018291850393775597208662, −10.59224965301357593165731165309, −9.858142998421329586247097934271, −8.939160980827677868507278916012, −8.123563353287430431527244767704, −6.82925488896532225918105148259, −5.55949156749524866516973330256, −4.96393915148140038987392594617, −3.09087052806274083177897050892, −1.97046487876769149429060984042,
1.51776662830399080839565989131, 2.81956669829911059749810591717, 4.79633566736593300562678079410, 5.29585085309259754788388190565, 6.86996572792364103692108380625, 7.918584000673338128116486956810, 8.386293588671690289914134759928, 10.10265259806200670970660982133, 10.36254226096584407145296461251, 11.37282595454449196833953317961