L(s) = 1 | + (0.866 + 1.11i)2-s − 1.73·3-s + (−0.500 + 1.93i)4-s + 2.23i·5-s + (−1.49 − 1.93i)6-s + (−2.59 + 1.11i)8-s + 2.99·9-s + (−2.50 + 1.93i)10-s + (0.866 − 3.35i)12-s − 3.87i·15-s + (−3.5 − 1.93i)16-s + 6.92·17-s + (2.59 + 3.35i)18-s + 4·19-s + (−4.33 − 1.11i)20-s + ⋯ |
L(s) = 1 | + (0.612 + 0.790i)2-s − 1.00·3-s + (−0.250 + 0.968i)4-s + 0.999i·5-s + (−0.612 − 0.790i)6-s + (−0.918 + 0.395i)8-s + 0.999·9-s + (−0.790 + 0.612i)10-s + (0.250 − 0.968i)12-s − 1.00i·15-s + (−0.875 − 0.484i)16-s + 1.68·17-s + (0.612 + 0.790i)18-s + 0.917·19-s + (−0.968 − 0.250i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.571002 + 0.867349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.571002 + 0.867349i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 1.11i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8.94iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7.74iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 15.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−14.07583105127777397105014190287, −12.67563606505328238744607703081, −11.96579569717191887322583573782, −10.85671098204716585909342961535, −9.772644300205767193251000508362, −7.955335915614753098599944535934, −6.93955839577251695221461704680, −6.05532064796848729882870449626, −4.90396513482860089923114534067, −3.30604674302533284706138136287,
1.24871830823032357325157849446, 3.78898483280470033170864145423, 5.19345208285814782747346326183, 5.79089491580389829527964502812, 7.60429980742977750577878436097, 9.426464548989252731432720387646, 10.09576733678791506577609402592, 11.54758214659964146686361762609, 11.95630402599732234585551874834, 12.97161248345628890331946702714