L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (1.73 − 2i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (2.5 − 4.33i)11-s + (−0.866 + 0.499i)12-s + 5i·13-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + (0.866 + 0.499i)18-s + (−3.5 − 6.06i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.753 − 1.30i)11-s + (−0.249 + 0.144i)12-s + 1.38i·13-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + (0.204 + 0.117i)18-s + (−0.802 − 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.080612481\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.080612481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + (9.52 - 5.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.836230913242713066857409131356, −8.881109539428411400235510862392, −7.965523070310263128836411578620, −6.82497705710978033604737071180, −6.35651595112453648970859603778, −5.26837170203719485089935673111, −4.34395517083850080423882480809, −3.59178548527415140290518200655, −2.03869198244360550765590247557, −0.881726754073095681531316022913,
1.64621132292252629200048457183, 3.04098226914475737464000653999, 4.22769619086154226136444841032, 5.01310478053945939279262613111, 5.77387077129675827100604285956, 6.52975980915240054040325065528, 7.71362690989653334218789450068, 8.203942943337333692580195073412, 9.457899527853407590647398402883, 10.13206576430589535984812050184