L(s) = 1 | + 2.30i·2-s + (−0.921 − 0.921i)3-s − 3.30·4-s + (−1.62 − 1.62i)5-s + (2.12 − 2.12i)6-s + (0.214 − 0.214i)7-s − 3.00i·8-s − 1.30i·9-s + (3.74 − 3.74i)10-s + (2.12 − 2.12i)11-s + (3.04 + 3.04i)12-s + 3.30·13-s + (0.493 + 0.493i)14-s + 3i·15-s + 0.302·16-s + ⋯ |
L(s) = 1 | + 1.62i·2-s + (−0.531 − 0.531i)3-s − 1.65·4-s + (−0.728 − 0.728i)5-s + (0.866 − 0.866i)6-s + (0.0809 − 0.0809i)7-s − 1.06i·8-s − 0.434i·9-s + (1.18 − 1.18i)10-s + (0.639 − 0.639i)11-s + (0.878 + 0.878i)12-s + 0.916·13-s + (0.131 + 0.131i)14-s + 0.774i·15-s + 0.0756·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.776041 - 0.0475879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.776041 - 0.0475879i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 3 | \( 1 + (0.921 + 0.921i)T + 3iT^{2} \) |
| 5 | \( 1 + (1.62 + 1.62i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.214 + 0.214i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.12 + 2.12i)T - 11iT^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 19 | \( 1 + 5.90iT - 19T^{2} \) |
| 23 | \( 1 + (-1.62 + 1.62i)T - 23iT^{2} \) |
| 29 | \( 1 + (7.00 + 7.00i)T + 29iT^{2} \) |
| 31 | \( 1 + (-2.54 - 2.54i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.428 - 0.428i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.24 - 4.24i)T - 41iT^{2} \) |
| 43 | \( 1 + 2.39iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 2.09iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + (-6.23 + 6.23i)T - 61iT^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + (2.27 + 2.27i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.278 + 0.278i)T + 73iT^{2} \) |
| 79 | \( 1 + (-7.92 + 7.92i)T - 79iT^{2} \) |
| 83 | \( 1 + 2.51iT - 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 + (-7.71 - 7.71i)T + 97iT^{2} \) |
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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
−11.78853995596079153925759031561, −11.18918452866924144060568263700, −9.298664642189460568827923653355, −8.632708390699628781774367604389, −7.79452130087395223949628832233, −6.72271495860641107291319298142, −6.13185587054695289610042114041, −4.99166438260935372295329067012, −3.87438777188536064300634592030, −0.66469491669643164941383628440,
1.77431148468465594407927335305, 3.45531144562811560230955273727, 4.08995802411882154002554354026, 5.41203026112395116941722145693, 6.96041850108825243479115703198, 8.258906710180038090128875705642, 9.495252297284421128755928071754, 10.29444590449976926502907510242, 11.08826561699529803781413280083, 11.46705168899192555656767006317