L(s) = 1 | + (−2.37 + 2.37i)5-s − 3.64i·7-s + (0.841 − 0.841i)11-s + (−2.64 − 2.64i)13-s + 3.06·17-s + (−1.64 − 1.64i)19-s − 7.82i·23-s − 6.29i·25-s + (−0.692 − 0.692i)29-s + 0.354·31-s + (8.66 + 8.66i)35-s + (4.64 − 4.64i)37-s − 6.43i·41-s + (5.64 − 5.64i)43-s − 11.1·47-s + ⋯ |
L(s) = 1 | + (−1.06 + 1.06i)5-s − 1.37i·7-s + (0.253 − 0.253i)11-s + (−0.733 − 0.733i)13-s + 0.744·17-s + (−0.377 − 0.377i)19-s − 1.63i·23-s − 1.25i·25-s + (−0.128 − 0.128i)29-s + 0.0636·31-s + (1.46 + 1.46i)35-s + (0.763 − 0.763i)37-s − 1.00i·41-s + (0.860 − 0.860i)43-s − 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.588158 - 0.606881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.588158 - 0.606881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.37 - 2.37i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.64iT - 7T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.841i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.64 + 2.64i)T + 13iT^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + (1.64 + 1.64i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.82iT - 23T^{2} \) |
| 29 | \( 1 + (0.692 + 0.692i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.354T + 31T^{2} \) |
| 37 | \( 1 + (-4.64 + 4.64i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.43iT - 41T^{2} \) |
| 43 | \( 1 + (-5.64 + 5.64i)T - 43iT^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + (5.44 - 5.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.82 - 7.82i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.64 - 4.64i)T + 61iT^{2} \) |
| 67 | \( 1 + (4 + 4i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 7.29iT - 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + (0.841 + 0.841i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.50iT - 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−10.61771265506397281220977271671, −9.957823710598590054596532040421, −8.541672215720855825554491715371, −7.51135867039395039852561895104, −7.22783377862116832991194847954, −6.13645382596370185449604661913, −4.58086570027083465785087417541, −3.75768049762624845140863804206, −2.77099561056716196288583004643, −0.48855585752752740098526550482,
1.65900566504633234390182008241, 3.24700311879794862244579865170, 4.49900147809910007647005607967, 5.23565260906944842262756020221, 6.33834987775895761153973079237, 7.70045840242602889049956372492, 8.224734465828751966175511731478, 9.274162075520657652718794397125, 9.678907734284198788217785249084, 11.39866908025891911834449955259