L(s) = 1 | + (0.747 + 0.431i)2-s + (−0.627 − 1.08i)4-s + (0.0476 + 2.23i)5-s + 0.566i·7-s − 2.80i·8-s + (−0.928 + 1.69i)10-s + 1.91·11-s + (−0.168 + 0.0972i)13-s + (−0.244 + 0.423i)14-s + (−0.0438 + 0.0760i)16-s + (4.58 + 2.64i)17-s + (2.36 + 3.65i)19-s + (2.40 − 1.45i)20-s + (1.42 + 0.824i)22-s + (2.92 − 1.68i)23-s + ⋯ |
L(s) = 1 | + (0.528 + 0.305i)2-s + (−0.313 − 0.543i)4-s + (0.0213 + 0.999i)5-s + 0.214i·7-s − 0.993i·8-s + (−0.293 + 0.534i)10-s + 0.576·11-s + (−0.0467 + 0.0269i)13-s + (−0.0653 + 0.113i)14-s + (−0.0109 + 0.0190i)16-s + (1.11 + 0.642i)17-s + (0.543 + 0.839i)19-s + (0.536 − 0.325i)20-s + (0.304 + 0.175i)22-s + (0.609 − 0.351i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81089 + 0.774793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81089 + 0.774793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.0476 - 2.23i)T \) |
| 19 | \( 1 + (-2.36 - 3.65i)T \) |
good | 2 | \( 1 + (-0.747 - 0.431i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 - 0.566iT - 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + (0.168 - 0.0972i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.58 - 2.64i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.92 + 1.68i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.36 - 7.56i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 0.955iT - 37T^{2} \) |
| 41 | \( 1 + (-5.02 + 8.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.27 + 2.46i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.65 - 4.41i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.10 - 4.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.85 - 3.21i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.75 - 3.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.50 + 2.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.59 + 4.49i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.45 - 4.30i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.31 + 5.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.51iT - 83T^{2} \) |
| 89 | \( 1 + (1.68 + 2.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 + 7.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.26559821522325428227827409889, −9.637905711129668643780893700101, −8.618381222230851886599178559092, −7.50770549437324376365139390054, −6.62545049676923803319192391415, −5.97684234378048338750123544877, −5.09845012852547660866181017840, −3.92055776184960951108490788917, −3.06493475997924747764611791837, −1.37371532421171773173121824289,
0.987357862996166031487783090832, 2.68658290892472637134242952658, 3.74044212783345727943274057268, 4.71464612528221052077952227245, 5.26178083232976903067068380929, 6.53884329952143064202705257797, 7.78295782164131559137755784160, 8.274771907871089101299146773215, 9.357626878390318871792589518630, 9.796247458164800074778304073899