| L(s) = 1 | + (1 − i)2-s − 2i·4-s − 5-s − 3i·7-s + (−2 − 2i)8-s + (−1 + i)10-s − 2·11-s + (−3 + 2i)13-s + (−3 − 3i)14-s − 4·16-s − 3·17-s + 2i·20-s + (−2 + 2i)22-s + 6·23-s − 4·25-s + (−1 + 5i)26-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s − 0.447·5-s − 1.13i·7-s + (−0.707 − 0.707i)8-s + (−0.316 + 0.316i)10-s − 0.603·11-s + (−0.832 + 0.554i)13-s + (−0.801 − 0.801i)14-s − 16-s − 0.727·17-s + 0.447i·20-s + (−0.426 + 0.426i)22-s + 1.25·23-s − 0.800·25-s + (−0.196 + 0.980i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.116245 + 1.17396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116245 + 1.17396i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−9.838685379266236219076443981276, −9.051255729462906967283285122444, −7.71697366843732888175956990579, −7.06301245441252675334246236615, −6.05299920579516037395756635088, −4.79578636324188286305997361737, −4.29353286187484130695075389112, −3.23257123115486763657558912311, −2.04571798562098403577891944973, −0.41005069647730060555325328768,
2.43280148417257110668833704133, 3.22520343473956141973312925965, 4.59174313634459510366590335032, 5.25323210754301293279961185377, 6.09090916221809287640824532885, 7.13176885768600318265574262979, 7.83881314973015279409779516984, 8.677899163356811975775206407300, 9.367058236941436946838956909401, 10.65612740236107695790796582103
