L(s) = 1 | + (−0.370 − 1.69i)3-s − 5-s + 0.0825i·7-s + (−2.72 + 1.25i)9-s + 1.67·11-s + 4.50i·13-s + (0.370 + 1.69i)15-s − 1.74i·17-s − 1.02·19-s + (0.139 − 0.0305i)21-s − 3.31i·23-s + 25-s + (3.13 + 4.14i)27-s + 3.47i·29-s − 4.73i·31-s + ⋯ |
L(s) = 1 | + (−0.213 − 0.976i)3-s − 0.447·5-s + 0.0312i·7-s + (−0.908 + 0.417i)9-s + 0.503·11-s + 1.24i·13-s + (0.0956 + 0.436i)15-s − 0.422i·17-s − 0.234·19-s + (0.0304 − 0.00667i)21-s − 0.692i·23-s + 0.200·25-s + (0.602 + 0.798i)27-s + 0.645i·29-s − 0.850i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.035247933\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035247933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.370 + 1.69i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (5.31 + 6.22i)T \) |
good | 7 | \( 1 - 0.0825iT - 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.50iT - 13T^{2} \) |
| 17 | \( 1 + 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 - 3.47iT - 29T^{2} \) |
| 31 | \( 1 + 4.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 + 12.7iT - 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 + 7.15iT - 59T^{2} \) |
| 61 | \( 1 - 1.24iT - 61T^{2} \) |
| 71 | \( 1 + 8.32iT - 71T^{2} \) |
| 73 | \( 1 - 9.54T + 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 2.48iT - 83T^{2} \) |
| 89 | \( 1 + 2.00iT - 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
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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
−8.147425187994627134889097584602, −7.32893685290679392889009352605, −6.74444218694566955866140808886, −6.24577472277893229766830517492, −5.22227230170035211374502431134, −4.42587018142387192599322413958, −3.51285765768969029553740635200, −2.41899688158633775502895606361, −1.59275645476910949893809802066, −0.36238209072098645433033418805,
1.01009449174626467924365625990, 2.59506893951629465374967485449, 3.49020488848017660676201228027, 4.05207172990083746971704266649, 4.94392509234702216021962494594, 5.65261953005223151311700401358, 6.32759471088406017369283425354, 7.32747526418309782636974386240, 8.089780123607092799971711968943, 8.747152326935005483196577061938