| L(s) = 1 | + (−0.5 + 0.866i)3-s − i·5-s + (−4.33 + 2.5i)7-s + (1 + 1.73i)9-s + (−1.73 − i)11-s + (0.866 + 0.5i)15-s + (−1.5 − 2.59i)17-s + (−1.73 + i)19-s − 5i·21-s + (2 − 3.46i)23-s + 4·25-s − 5·27-s + (3 − 5.19i)29-s − 4i·31-s + (1.73 − 0.999i)33-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s − 0.447i·5-s + (−1.63 + 0.944i)7-s + (0.333 + 0.577i)9-s + (−0.522 − 0.301i)11-s + (0.223 + 0.129i)15-s + (−0.363 − 0.630i)17-s + (−0.397 + 0.229i)19-s − 1.09i·21-s + (0.417 − 0.722i)23-s + 0.800·25-s − 0.962·27-s + (0.557 − 0.964i)29-s − 0.718i·31-s + (0.301 − 0.174i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0771 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0771 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5324249698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5324249698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + (4.33 - 2.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.73 - i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.92 + 4i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + (5.19 - 3i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.06 + 3.5i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + (8.66 + 5i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.500472046488218777798812338431, −8.766173611373717048673526967359, −7.931905485928080433796206254554, −6.72974846728988621204162290515, −6.08462361058427509271749838180, −5.17760118262561966666870864745, −4.42579875972422036329490191242, −3.17015773683181319454697057457, −2.33340675723721922095429587519, −0.24680720501100630275765823600,
1.17869688948972168068245284548, 2.85385374284023703585571842619, 3.58903033016519027000351151858, 4.63167397604240893917344727222, 6.00528044251770672111656043103, 6.70025017608567996738905910853, 7.01667040069550958593972002565, 7.976690318795666859585653622294, 9.264591493723696602939310583613, 9.740893962891620478694304481628
