L(s) = 1 | + (1.86 + 0.5i)3-s + (−1 − 2i)5-s + (2.23 − 3.86i)7-s + (0.633 + 0.366i)9-s + (−2.86 − 0.767i)11-s + (3 + 2i)13-s + (−0.866 − 4.23i)15-s + (0.866 + 3.23i)17-s + (1.13 + 4.23i)19-s + (6.09 − 6.09i)21-s + (−1.86 + 6.96i)23-s + (−3 + 4i)25-s + (−3.09 − 3.09i)27-s + (1.5 − 0.866i)29-s + (5.19 + 5.19i)31-s + ⋯ |
L(s) = 1 | + (1.07 + 0.288i)3-s + (−0.447 − 0.894i)5-s + (0.843 − 1.46i)7-s + (0.211 + 0.122i)9-s + (−0.864 − 0.231i)11-s + (0.832 + 0.554i)13-s + (−0.223 − 1.09i)15-s + (0.210 + 0.783i)17-s + (0.260 + 0.970i)19-s + (1.33 − 1.33i)21-s + (−0.389 + 1.45i)23-s + (−0.600 + 0.800i)25-s + (−0.596 − 0.596i)27-s + (0.278 − 0.160i)29-s + (0.933 + 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60710 - 0.479024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60710 - 0.479024i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 3 | \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (-2.23 + 3.86i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.86 + 0.767i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 3.23i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 4.23i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (1.86 - 6.96i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.19 - 5.19i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.23 + 7.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.669 + 2.5i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.59 + 0.964i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + (4.46 - 4.46i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.33 + 1.69i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.6 - 7.86i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.59 - 1.23i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 1.07iT - 73T^{2} \) |
| 79 | \( 1 - 7.46iT - 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + (0.794 - 2.96i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.69 - 2.13i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.86701497832908435500959430038, −10.85640098871989739182645696355, −9.962423034625890182360022266476, −8.757723796705884927624622649650, −8.097190214778718078640450167220, −7.45762509760061540198987733183, −5.60760672594298418688906956480, −4.21770592090254312313077881580, −3.59446630296290096976854578788, −1.47360264029622721508204285114,
2.43032236749597662862668088796, 2.96304883255103036573203004353, 4.80972659012944149074876397327, 6.08927578737864437354592114204, 7.47796226374428617936103785903, 8.253779384272495290365209100204, 8.822929390202160325820307593568, 10.18276678814458206686959655026, 11.25125199180344965795413134140, 11.96565310257565593193952640416