L(s) = 1 | + (−0.112 − 0.0302i)2-s + (2.25 − 1.29i)3-s + (−1.72 − 0.993i)4-s + (1.24 + 1.24i)5-s + (−0.293 + 0.0785i)6-s + (−2.63 − 0.206i)7-s + (0.329 + 0.329i)8-s + (1.87 − 3.25i)9-s + (−0.103 − 0.178i)10-s + (−0.506 + 1.89i)11-s − 5.16·12-s + (1.85 + 3.09i)13-s + (0.291 + 0.102i)14-s + (4.43 + 1.18i)15-s + (1.95 + 3.39i)16-s + (2.13 − 3.70i)17-s + ⋯ |
L(s) = 1 | + (−0.0797 − 0.0213i)2-s + (1.29 − 0.750i)3-s + (−0.860 − 0.496i)4-s + (0.558 + 0.558i)5-s + (−0.119 + 0.0320i)6-s + (−0.996 − 0.0779i)7-s + (0.116 + 0.116i)8-s + (0.625 − 1.08i)9-s + (−0.0325 − 0.0564i)10-s + (−0.152 + 0.570i)11-s − 1.49·12-s + (0.515 + 0.857i)13-s + (0.0778 + 0.0275i)14-s + (1.14 + 0.306i)15-s + (0.489 + 0.848i)16-s + (0.518 − 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09884 - 0.345799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09884 - 0.345799i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.63 + 0.206i)T \) |
| 13 | \( 1 + (-1.85 - 3.09i)T \) |
good | 2 | \( 1 + (0.112 + 0.0302i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (-2.25 + 1.29i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.24 - 1.24i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.506 - 1.89i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.13 + 3.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.12 - 1.10i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.53 - 3.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.02 - 3.02i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.732 - 2.73i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.94 + 11.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.68 + 4.68i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + (-0.436 - 1.62i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (2.66 + 1.54i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0190 + 0.00510i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.698 - 0.698i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.93T + 79T^{2} \) |
| 83 | \( 1 + (9.87 + 9.87i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.76 - 2.07i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (14.2 - 3.82i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80010591585430230931469204094, −13.45800661860562715137023791302, −12.24237237764552865981001229602, −10.24573772177516947473347994055, −9.535095396075641068879585863140, −8.581278053175267902452658637357, −7.22538530586518331718293363237, −6.03650395405529175532830368873, −3.87674659721919881274481099324, −2.19299471352711095509131907157,
3.08503705595172998221242456262, 4.13772131562528783521420535183, 5.84155019237828579655034125381, 8.052059925127308316242178111356, 8.742589136418982054494366168100, 9.570117778330969421436528436147, 10.43260341144690577824830332438, 12.69797041005757625667567986745, 13.15725315906503540727585668470, 14.10108825573344828153179575995