| L(s) = 1 | + 0.359·3-s + (0.565 − 2.16i)5-s + i·7-s − 2.87·9-s + 1.56i·11-s − 6.61·13-s + (0.203 − 0.776i)15-s − 2.81i·17-s − 5.37i·19-s + 0.359i·21-s + 5.85i·23-s + (−4.35 − 2.44i)25-s − 2.10·27-s − 6.75i·29-s − 9.18·31-s + ⋯ |
| L(s) = 1 | + 0.207·3-s + (0.253 − 0.967i)5-s + 0.377i·7-s − 0.957·9-s + 0.472i·11-s − 1.83·13-s + (0.0524 − 0.200i)15-s − 0.683i·17-s − 1.23i·19-s + 0.0783i·21-s + 1.22i·23-s + (−0.871 − 0.489i)25-s − 0.405·27-s − 1.25i·29-s − 1.65·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4650761529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4650761529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.565 + 2.16i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 0.359T + 3T^{2} \) |
| 11 | \( 1 - 1.56iT - 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 2.81iT - 17T^{2} \) |
| 19 | \( 1 + 5.37iT - 19T^{2} \) |
| 23 | \( 1 - 5.85iT - 23T^{2} \) |
| 29 | \( 1 + 6.75iT - 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 7.93T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 - 5.24iT - 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 3.87iT - 59T^{2} \) |
| 61 | \( 1 - 4.69iT - 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.67iT - 73T^{2} \) |
| 79 | \( 1 - 8.46T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 + 4.21T + 89T^{2} \) |
| 97 | \( 1 - 5.89iT - 97T^{2} \) |
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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
−9.346993155500021710423801545463, −8.884754629795982937665483750397, −7.75652844111204427254970322958, −7.18259801533620486906440695618, −5.78391759281728121944976192561, −5.18388807438647558624852996366, −4.39719571788497254146190431716, −2.88012436970926492590245345003, −2.04613305098160515119005388193, −0.17773753770669591491579010501,
2.04347783997167304726673911533, 2.97645147509086220875684152961, 3.87482818916933175362297812470, 5.22933779839070631855630417497, 6.00149998024187669257595868815, 6.96936472092424888083517537137, 7.65155971619910769145228146641, 8.552922073867425561051813376445, 9.437911049705906780121457551626, 10.44124047184982219060764459684
