| L(s) = 1 | + (0.704 − 1.22i)2-s + (0.793 − 0.608i)3-s + (−1.00 − 1.72i)4-s + (−2.36 + 3.07i)5-s + (−0.187 − 1.40i)6-s + (0.609 + 2.57i)7-s + (−2.82 + 0.0203i)8-s + (0.258 − 0.965i)9-s + (2.11 + 5.06i)10-s + (−0.0301 + 0.228i)11-s + (−1.85 − 0.756i)12-s + (−2.01 + 4.86i)13-s + (3.58 + 1.06i)14-s + 3.87i·15-s + (−1.96 + 3.48i)16-s + (6.56 + 3.79i)17-s + ⋯ |
| L(s) = 1 | + (0.497 − 0.867i)2-s + (0.458 − 0.351i)3-s + (−0.504 − 0.863i)4-s + (−1.05 + 1.37i)5-s + (−0.0767 − 0.572i)6-s + (0.230 + 0.973i)7-s + (−0.999 + 0.00719i)8-s + (0.0862 − 0.321i)9-s + (0.667 + 1.60i)10-s + (−0.00908 + 0.0690i)11-s + (−0.534 − 0.218i)12-s + (−0.559 + 1.34i)13-s + (0.958 + 0.284i)14-s + 1.00i·15-s + (−0.491 + 0.870i)16-s + (1.59 + 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34344 + 0.444647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34344 + 0.444647i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.704 + 1.22i)T \) |
| 3 | \( 1 + (-0.793 + 0.608i)T \) |
| 7 | \( 1 + (-0.609 - 2.57i)T \) |
| good | 5 | \( 1 + (2.36 - 3.07i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (0.0301 - 0.228i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (2.01 - 4.86i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-6.56 - 3.79i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.91 - 0.647i)T + (18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.54 + 5.77i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.95 - 2.05i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (1.38 - 2.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.03 - 9.16i)T + (-9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (1.67 + 1.67i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.81 + 1.16i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.600 + 0.346i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.597 + 4.53i)T + (-51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (0.0847 + 0.0111i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.0708 + 0.537i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-6.38 + 4.89i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (2.51 - 2.51i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.185 - 0.0496i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.13 - 1.23i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.05 + 2.53i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-7.35 - 1.97i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−10.64896528121530567369277584151, −10.06953493776553863606185064994, −8.796366244965972735249020241940, −8.170395876322440643620696282141, −6.88689916460207349698502802338, −6.26937577268738154352474034359, −4.80706823449592657362284036193, −3.73942565516232134215646830526, −2.89362423138600666271982685881, −1.93320874724448671324013511403,
0.61592470016111296506879610197, 3.23696318798470226717617260412, 4.04798953634587370873933070692, 4.90185037820806754056236984765, 5.54936670504747145875183775988, 7.34342798353470468222187828686, 7.70976787881047768002300036998, 8.391470273394484329738635626854, 9.281991204630890677343220206949, 10.24717655674905076074192180772
