L(s) = 1 | + (0.635 + 1.26i)2-s + (0.707 − 0.707i)3-s + (−1.19 + 1.60i)4-s + (−2.68 − 2.68i)5-s + (1.34 + 0.443i)6-s + 2.15i·7-s + (−2.78 − 0.484i)8-s − 1.00i·9-s + (1.68 − 5.09i)10-s + (1.79 + 1.79i)11-s + (0.292 + 1.97i)12-s + (1.38 − 1.38i)13-s + (−2.72 + 1.37i)14-s − 3.79·15-s + (−1.15 − 3.82i)16-s − 0.224·17-s + ⋯ |
L(s) = 1 | + (0.449 + 0.893i)2-s + (0.408 − 0.408i)3-s + (−0.595 + 0.803i)4-s + (−1.20 − 1.20i)5-s + (0.548 + 0.181i)6-s + 0.816i·7-s + (−0.985 − 0.171i)8-s − 0.333i·9-s + (0.533 − 1.61i)10-s + (0.542 + 0.542i)11-s + (0.0845 + 0.571i)12-s + (0.383 − 0.383i)13-s + (−0.728 + 0.366i)14-s − 0.980·15-s + (−0.289 − 0.957i)16-s − 0.0545·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.871540 + 0.311560i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.871540 + 0.311560i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.635 - 1.26i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (2.68 + 2.68i)T + 5iT^{2} \) |
| 7 | \( 1 - 2.15iT - 7T^{2} \) |
| 11 | \( 1 + (-1.79 - 1.79i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.224T + 17T^{2} \) |
| 19 | \( 1 + (-0.158 + 0.158i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (1.85 - 1.85i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + (3.66 + 3.66i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.88iT - 41T^{2} \) |
| 43 | \( 1 + (7.75 + 7.75i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4 - 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.98 + 5.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.4 - 10.4i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.31iT - 71T^{2} \) |
| 73 | \( 1 + 5.97iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + (10.1 - 10.1i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.42iT - 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−15.60478899011562921687979644080, −15.02615018984691346279224474452, −13.52401554393183182785110398502, −12.43031386523642399712733001277, −11.88643337552233925390382200980, −9.086993693289973544211052872954, −8.381293715168107984791390773270, −7.20809824324198859285104900790, −5.37842133309248723250807203915, −3.86179068547023561844984227152,
3.26001583815327479995642235752, 4.23338644021661599711126263758, 6.67926877056311355946931200854, 8.340574390110295203834746480786, 10.01882682873606116854011464606, 11.02654065873239224617703655017, 11.73266118695565942172873152568, 13.45950518284363879693775285106, 14.41983232478527245787039181785, 15.15685271547203927047634424941