L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 + 0.707i)5-s + (4.26 − 1.14i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−5.38 − 1.44i)11-s + (−3.06 − 1.90i)13-s − 4.41i·14-s + (0.500 + 0.866i)16-s + (0.909 − 1.57i)17-s + (−1.88 − 7.02i)19-s + (−0.258 − 0.965i)20-s + (−2.78 + 4.82i)22-s + (2.07 + 3.59i)23-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.316 + 0.316i)5-s + (1.61 − 0.432i)7-s + (−0.249 + 0.249i)8-s + (0.273 − 0.158i)10-s + (−1.62 − 0.434i)11-s + (−0.849 − 0.527i)13-s − 1.18i·14-s + (0.125 + 0.216i)16-s + (0.220 − 0.382i)17-s + (−0.431 − 1.61i)19-s + (−0.0578 − 0.215i)20-s + (−0.594 + 1.02i)22-s + (0.432 + 0.749i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.720709686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.720709686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (3.06 + 1.90i)T \) |
good | 7 | \( 1 + (-4.26 + 1.14i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (5.38 + 1.44i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.909 + 1.57i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.88 + 7.02i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.07 - 3.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.49 + 3.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.75 + 3.75i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.09 + 7.81i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.03 + 3.86i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.26 - 3.61i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 - 6.08i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.04iT - 53T^{2} \) |
| 59 | \( 1 + (0.102 + 0.382i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.26 - 5.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.04 - 0.815i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (13.8 - 3.71i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (1.33 + 1.33i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + (3.28 + 3.28i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.23 - 1.40i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.48 - 5.53i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−9.754318512659115274850395863200, −8.705307958367817728355757179271, −7.81301491357702378624063809662, −7.33861915364961126179388275801, −5.81687861769622179404153649678, −4.99663523986962200189334947803, −4.49958214953271969187779141203, −2.87867135340135121310409325347, −2.29459307323215143323385804812, −0.70778869028304561295349966267,
1.64173951228021254811735738534, 2.73082273879372331430185486825, 4.49738307750440398508633272987, 4.91709400488712957226082137808, 5.65157245753863401627481264406, 6.71848670699211251167296194999, 7.967792574345043432763714082295, 8.028180547689130853998516239564, 8.957681913771617114111827003428, 10.17215918057078414878806884131