L(s) = 1 | − 2.11·2-s + i·3-s + 2.47·4-s + 1.98·5-s − 2.11i·6-s + (−2.62 − 0.302i)7-s − 1.00·8-s − 9-s − 4.20·10-s − 2.93i·11-s + 2.47i·12-s − 4.75i·13-s + (5.55 + 0.638i)14-s + 1.98i·15-s − 2.83·16-s − 0.444·17-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 0.577i·3-s + 1.23·4-s + 0.889·5-s − 0.863i·6-s + (−0.993 − 0.114i)7-s − 0.353·8-s − 0.333·9-s − 1.33·10-s − 0.883i·11-s + 0.713i·12-s − 1.31i·13-s + (1.48 + 0.170i)14-s + 0.513i·15-s − 0.707·16-s − 0.107·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.324474 - 0.298588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.324474 - 0.298588i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.62 + 0.302i)T \) |
| 23 | \( 1 + (-4.70 + 0.940i)T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 11 | \( 1 + 2.93iT - 11T^{2} \) |
| 13 | \( 1 + 4.75iT - 13T^{2} \) |
| 17 | \( 1 + 0.444T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 6.35iT - 31T^{2} \) |
| 37 | \( 1 + 6.69iT - 37T^{2} \) |
| 41 | \( 1 + 10.5iT - 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 5.66iT - 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 - 6.39iT - 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 + 9.73iT - 67T^{2} \) |
| 71 | \( 1 - 8.75T + 71T^{2} \) |
| 73 | \( 1 - 0.926iT - 73T^{2} \) |
| 79 | \( 1 - 8.86iT - 79T^{2} \) |
| 83 | \( 1 + 4.81T + 83T^{2} \) |
| 89 | \( 1 - 0.0511T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−10.61852756982900967923616585466, −9.866020536664119414924998023416, −8.979205232367181362298375329183, −8.552681798607522324838115939812, −7.26759034583948714663011294213, −6.26668225224520818725246693440, −5.36230378401750704956893652798, −3.60815073457334114195230687116, −2.32578866399791171734492496739, −0.42466584081625905436003491518,
1.58775319213468803849386458630, 2.50684068057011214115807020072, 4.49297340888333469216494495704, 6.25210836025236319684533930391, 6.66415222482631463470044869277, 7.63962196380475713871165601832, 8.738699298025780362853036623212, 9.518159155862584492500478538744, 9.844443208512583511135068054958, 10.96046009629797589959304157644