L(s) = 1 | + (−0.524 − 1.31i)2-s + (2.57 + 0.689i)3-s + (−1.45 + 1.37i)4-s + (0.222 − 2.22i)5-s + (−0.443 − 3.73i)6-s + (1.04 − 0.280i)7-s + (2.56 + 1.18i)8-s + (3.54 + 2.04i)9-s + (−3.03 + 0.874i)10-s + (0.213 − 0.123i)11-s + (−4.67 + 2.54i)12-s + (−3.26 − 1.52i)13-s + (−0.915 − 1.22i)14-s + (2.10 − 5.56i)15-s + (0.206 − 3.99i)16-s + (1.56 + 5.82i)17-s + ⋯ |
L(s) = 1 | + (−0.370 − 0.928i)2-s + (1.48 + 0.397i)3-s + (−0.725 + 0.688i)4-s + (0.0995 − 0.995i)5-s + (−0.180 − 1.52i)6-s + (0.395 − 0.105i)7-s + (0.908 + 0.418i)8-s + (1.18 + 0.681i)9-s + (−0.961 + 0.276i)10-s + (0.0643 − 0.0371i)11-s + (−1.35 + 0.733i)12-s + (−0.906 − 0.422i)13-s + (−0.244 − 0.327i)14-s + (0.543 − 1.43i)15-s + (0.0516 − 0.998i)16-s + (0.378 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35605 - 0.840026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35605 - 0.840026i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.524 + 1.31i)T \) |
| 5 | \( 1 + (-0.222 + 2.22i)T \) |
| 13 | \( 1 + (3.26 + 1.52i)T \) |
good | 3 | \( 1 + (-2.57 - 0.689i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 0.280i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.213 + 0.123i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 5.82i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.97 + 3.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.77 - 0.742i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.76 - 2.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.881iT - 31T^{2} \) |
| 37 | \( 1 + (4.95 + 1.32i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.89 - 10.2i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.517 + 1.93i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (6.04 + 6.04i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.436 + 0.436i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.04 - 8.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.71 - 4.70i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.36 - 12.5i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-8.10 - 4.68i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.41 + 3.41i)T + 73iT^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + (-10.8 + 10.8i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.31 + 0.762i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.34 + 12.4i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.88379548043106006696248609851, −10.63129765722575529585439812568, −9.693068258146650548397732970278, −9.047970656506363949101460353848, −8.266992869924163208439914636980, −7.57318270214342619555941830030, −5.14466984540556197446458680091, −4.13255390299926493227827078554, −2.99471295932384229966964910600, −1.62825674677798347811662276268,
2.08660739412238997345147870416, 3.46757591587227668818044542930, 5.08309442939475310658101030196, 6.60909987441765280520191395572, 7.50774548195082812568048145726, 7.927249054067070208129668300731, 9.311258414580277252853711200581, 9.652334570523891138642922183425, 10.99094791180273421853300420139, 12.35735328260408480888052218776