L(s) = 1 | + (−1.41 − 0.0941i)2-s + (0.416 + 1.68i)3-s + (1.98 + 0.265i)4-s + (−1.62 + 1.54i)5-s + (−0.429 − 2.41i)6-s + (−0.361 + 0.361i)7-s + (−2.77 − 0.561i)8-s + (−2.65 + 1.40i)9-s + (2.43 − 2.02i)10-s + 2.63·11-s + (0.378 + 3.44i)12-s + (−3.49 + 3.49i)13-s + (0.544 − 0.476i)14-s + (−3.26 − 2.08i)15-s + (3.85 + 1.05i)16-s + (3.61 + 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0665i)2-s + (0.240 + 0.970i)3-s + (0.991 + 0.132i)4-s + (−0.724 + 0.688i)5-s + (−0.175 − 0.984i)6-s + (−0.136 + 0.136i)7-s + (−0.980 − 0.198i)8-s + (−0.884 + 0.466i)9-s + (0.769 − 0.638i)10-s + 0.794·11-s + (0.109 + 0.994i)12-s + (−0.968 + 0.968i)13-s + (0.145 − 0.127i)14-s + (−0.842 − 0.538i)15-s + (0.964 + 0.263i)16-s + (0.876 + 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.387251 + 0.498141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.387251 + 0.498141i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0941i)T \) |
| 3 | \( 1 + (-0.416 - 1.68i)T \) |
| 5 | \( 1 + (1.62 - 1.54i)T \) |
good | 7 | \( 1 + (0.361 - 0.361i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + (3.49 - 3.49i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.61 - 3.61i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.672T + 19T^{2} \) |
| 23 | \( 1 + (-4.31 + 4.31i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.76iT - 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + (-2.82 - 2.82i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.10iT - 41T^{2} \) |
| 43 | \( 1 + (-7.57 + 7.57i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.987 - 0.987i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.646 - 0.646i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.92iT - 59T^{2} \) |
| 61 | \( 1 + 6.07iT - 61T^{2} \) |
| 67 | \( 1 + (-0.349 - 0.349i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.63iT - 71T^{2} \) |
| 73 | \( 1 + (11.3 + 11.3i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.07iT - 79T^{2} \) |
| 83 | \( 1 + (8.53 + 8.53i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + (0.660 - 0.660i)T - 97iT^{2} \) |
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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
−14.34360678780914511726600028935, −12.21445941320958123168003102452, −11.47506850509659962639056179081, −10.47297560658996592500510125165, −9.620896344291763283605748000566, −8.625488320212957248207440457502, −7.49397795140186322597517730287, −6.25821606753875046110332018125, −4.20011787946726073988815900667, −2.78755002876772390421098382058,
0.965620455948851374096445281244, 3.08818423879855636304220805831, 5.48994113794303824584628921110, 7.11822383306970411121674410420, 7.68370159013067655949604153459, 8.795208228456685411754524824753, 9.711764431371147060825411557029, 11.30335185492922899725571171089, 12.09231161285637708419789312588, 12.83856093100641506945180748673