L(s) = 1 | + (0.621 − 1.27i)2-s + (1.72 + 0.121i)3-s + (−1.22 − 1.57i)4-s + (−1 − 2i)5-s + (1.22 − 2.11i)6-s + 7-s + (−2.76 + 0.578i)8-s + (2.97 + 0.419i)9-s + (−3.16 + 0.0276i)10-s − 3.45·11-s + (−1.92 − 2.87i)12-s − 4.83i·13-s + (0.621 − 1.27i)14-s + (−1.48 − 3.57i)15-s + (−0.985 + 3.87i)16-s + 5.94·17-s + ⋯ |
L(s) = 1 | + (0.439 − 0.898i)2-s + (0.997 + 0.0700i)3-s + (−0.613 − 0.789i)4-s + (−0.447 − 0.894i)5-s + (0.501 − 0.865i)6-s + 0.377·7-s + (−0.978 + 0.204i)8-s + (0.990 + 0.139i)9-s + (−0.999 + 0.00874i)10-s − 1.04·11-s + (−0.557 − 0.830i)12-s − 1.34i·13-s + (0.166 − 0.339i)14-s + (−0.383 − 0.923i)15-s + (−0.246 + 0.969i)16-s + 1.44·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02712 - 1.76411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02712 - 1.76411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.621 + 1.27i)T \) |
| 3 | \( 1 + (-1.72 - 0.121i)T \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + 4.83iT - 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 1.08iT - 19T^{2} \) |
| 23 | \( 1 - 0.596iT - 23T^{2} \) |
| 29 | \( 1 + 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 9.56iT - 31T^{2} \) |
| 37 | \( 1 + 2.91iT - 37T^{2} \) |
| 41 | \( 1 - 6.91iT - 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 + 0.242iT - 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 5.71T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 0.949iT - 79T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 - 5.23iT - 89T^{2} \) |
| 97 | \( 1 - 3.30iT - 97T^{2} \) |
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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
−10.73220505646378285205242137261, −10.12282218955213530606301949898, −9.169567963213845749017905377214, −8.155102628622112196527076869032, −7.71661539354894540088534362189, −5.60432240505301863426965565246, −4.86928610945997956625856290697, −3.69056027283988201261252647864, −2.74167686633502892408289158918, −1.16218146294091971219121674886,
2.50291805213141537860239152466, 3.59897479534566530568646732256, 4.55848756151163637461578221710, 5.92371432770562274245099604649, 7.20336873735126772433456785456, 7.56304892131079093507807202159, 8.472096090711542371468066974578, 9.450668390120927789494013911501, 10.46700681416693300580359472472, 11.69405565070037014642948945738