| L(s) = 1 | + 3.27i·5-s + (−2.63 + 0.209i)7-s − 6.50·11-s + 7.27i·17-s + 4.35i·19-s − 8.71·23-s − 5.72·25-s + (−0.685 − 8.63i)35-s + 11.8·43-s − 2.72i·47-s + (6.91 − 1.10i)49-s − 21.3i·55-s − 10.8i·61-s + 16.0i·73-s + (17.1 − 1.36i)77-s + ⋯ |
| L(s) = 1 | + 1.46i·5-s + (−0.996 + 0.0791i)7-s − 1.96·11-s + 1.76i·17-s + 0.999i·19-s − 1.81·23-s − 1.14·25-s + (−0.115 − 1.45i)35-s + 1.80·43-s − 0.397i·47-s + (0.987 − 0.157i)49-s − 2.87i·55-s − 1.38i·61-s + 1.88i·73-s + (1.95 − 0.155i)77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1118794054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1118794054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.209i)T \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 - 3.27iT - 5T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.27iT - 17T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 2.72iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 16.0iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.684191884764704245818278166236, −7.84815702202853388186755460797, −7.57702819285540442094873779143, −6.50015327742270569479729818810, −6.04239257736752281111941002112, −5.49343565777563395542004367468, −4.07122095561051138460243903168, −3.50750174785511774819188366317, −2.64962642282764680250501931564, −2.03218476892242346660688628487,
0.04084124142431136516061048982, 0.73675386368232727167534285122, 2.33236319269032641597618614584, 2.90818876092883664288765609777, 4.14053167441187102198583230992, 4.84893659703139497911491424112, 5.43377444597293221657423672040, 6.08879415266355529837215314443, 7.24069595421292759340134257382, 7.70537118235592702767235746238
