L(s) = 1 | + 1.17·3-s − 2.23i·5-s + 2.64i·7-s − 1.62·9-s + 0.359·11-s − 5.64i·13-s − 2.62i·15-s + 2.03·17-s + 3.10i·21-s − 5.00·25-s − 5.42·27-s − 10.7i·29-s + 0.422·33-s + 5.91·35-s − 6.62i·39-s + ⋯ |
L(s) = 1 | + 0.677·3-s − 0.999i·5-s + 0.999i·7-s − 0.541·9-s + 0.108·11-s − 1.56i·13-s − 0.677i·15-s + 0.492·17-s + 0.677i·21-s − 1.00·25-s − 1.04·27-s − 1.99i·29-s + 0.0735·33-s + 0.999·35-s − 1.06i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.619079448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.619079448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 11 | \( 1 - 0.359T + 11T^{2} \) |
| 13 | \( 1 + 5.64iT - 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.7iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.744375249542391420339903064747, −8.055738974852846911457531683972, −7.78374181380832545103901354097, −6.19032949507995195030840344524, −5.61813059382757026733068944913, −4.96209299200242880808746499636, −3.75990483784600868881489456520, −2.88364872621146870024423186486, −2.00940489209134896270944975485, −0.49400813934140932831287076684,
1.51248108327283766873478656349, 2.62862892748131086822167204501, 3.50109752538708758135364633141, 4.12005866445836339115776576280, 5.29066289013654706949873108392, 6.44104489755460993021161180991, 6.96422625531639181221313527683, 7.63998084796023327697582829203, 8.476651632869139678925374581422, 9.302001003671649993622304437443