L(s) = 1 | + (0.333 + 0.333i)3-s + (−2.17 + 0.523i)5-s + (−0.707 + 0.707i)7-s − 2.77i·9-s − 3.81i·11-s + (2.82 − 2.82i)13-s + (−0.898 − 0.550i)15-s + (−2.97 − 2.97i)17-s + 5.82·19-s − 0.471·21-s + (6.07 + 6.07i)23-s + (4.45 − 2.27i)25-s + (1.92 − 1.92i)27-s − 9.62i·29-s + 3.98i·31-s + ⋯ |
L(s) = 1 | + (0.192 + 0.192i)3-s + (−0.972 + 0.233i)5-s + (−0.267 + 0.267i)7-s − 0.925i·9-s − 1.15i·11-s + (0.783 − 0.783i)13-s + (−0.232 − 0.142i)15-s + (−0.720 − 0.720i)17-s + 1.33·19-s − 0.102·21-s + (1.26 + 1.26i)23-s + (0.890 − 0.454i)25-s + (0.370 − 0.370i)27-s − 1.78i·29-s + 0.716i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00903 - 0.579693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00903 - 0.579693i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.17 - 0.523i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.333 - 0.333i)T + 3iT^{2} \) |
| 11 | \( 1 + 3.81iT - 11T^{2} \) |
| 13 | \( 1 + (-2.82 + 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.97 + 2.97i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 + (-6.07 - 6.07i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.62iT - 29T^{2} \) |
| 31 | \( 1 - 3.98iT - 31T^{2} \) |
| 37 | \( 1 + (-1.50 - 1.50i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 + (7.59 + 7.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.02 - 4.02i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.08 + 2.08i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 6.60T + 61T^{2} \) |
| 67 | \( 1 + (1.76 - 1.76i)T - 67iT^{2} \) |
| 71 | \( 1 + 16.3iT - 71T^{2} \) |
| 73 | \( 1 + (-7.53 + 7.53i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.84T + 79T^{2} \) |
| 83 | \( 1 + (-0.140 - 0.140i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.64iT - 89T^{2} \) |
| 97 | \( 1 + (-7.77 - 7.77i)T + 97iT^{2} \) |
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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
−10.75996119029934909956066952940, −9.610422902978563888211088590911, −8.837115727295295115226720632018, −8.057716811340676702202255816390, −7.03516517508047387099897908749, −6.09163188050209775492561415362, −4.98289452770041646303649608494, −3.37345549507477044040434765935, −3.28258647812758374785276464434, −0.70938163066434369768598684967,
1.56710619892905845043860586862, 3.13923578954496264873737052676, 4.34606563781672878727665644257, 5.05970159775419987334864427863, 6.75310422261496117936198778940, 7.23203732478091354204121716690, 8.301812515190941883944212853938, 8.943284827850514698675050178217, 10.11461461455565516423894844083, 11.00781376374518086544581549134