L(s) = 1 | − 1.49·3-s − 1.37i·5-s + i·7-s − 0.776·9-s + i·11-s + (−2.93 + 2.09i)13-s + 2.05i·15-s + 0.801·17-s − 0.0452i·19-s − 1.49i·21-s + 6.18·23-s + 3.09·25-s + 5.63·27-s − 7.84·29-s + 2.31i·31-s + ⋯ |
L(s) = 1 | − 0.860·3-s − 0.616i·5-s + 0.377i·7-s − 0.258·9-s + 0.301i·11-s + (−0.814 + 0.580i)13-s + 0.530i·15-s + 0.194·17-s − 0.0103i·19-s − 0.325i·21-s + 1.29·23-s + 0.619·25-s + 1.08·27-s − 1.45·29-s + 0.415i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2947841554\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2947841554\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (2.93 - 2.09i)T \) |
good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 1.37iT - 5T^{2} \) |
| 17 | \( 1 - 0.801T + 17T^{2} \) |
| 19 | \( 1 + 0.0452iT - 19T^{2} \) |
| 23 | \( 1 - 6.18T + 23T^{2} \) |
| 29 | \( 1 + 7.84T + 29T^{2} \) |
| 31 | \( 1 - 2.31iT - 31T^{2} \) |
| 37 | \( 1 - 3.74iT - 37T^{2} \) |
| 41 | \( 1 + 2.15iT - 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + 1.91iT - 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 8.56iT - 59T^{2} \) |
| 61 | \( 1 - 7.82T + 61T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 - 1.36iT - 71T^{2} \) |
| 73 | \( 1 - 6.13iT - 73T^{2} \) |
| 79 | \( 1 + 8.88T + 79T^{2} \) |
| 83 | \( 1 + 4.20iT - 83T^{2} \) |
| 89 | \( 1 + 9.55iT - 89T^{2} \) |
| 97 | \( 1 + 5.59iT - 97T^{2} \) |
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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
−8.265786236476556399412199237414, −7.22878224894064328560715795574, −6.71147242286008199166989362965, −5.79480033754149987579268861269, −5.04316370619114537452642151466, −4.79085572233594116325232146516, −3.53263267605875061782440769837, −2.50158385399594058713860829691, −1.38656598647225187028721270466, −0.11279152575154912445327550592,
1.03507530715521609773207958220, 2.54319799900918159860890906502, 3.22549365077890596176950980882, 4.26132650974639190590734615274, 5.29225017716888373490550931638, 5.60540497251952153382611506365, 6.63080676193911000923184493607, 7.12295483125242381866886386731, 7.85456287017832552784831790915, 8.749870684810494167296799866298