L(s) = 1 | + (−0.756 − 0.756i)3-s + (−8.22 + 8.22i)5-s − 2.67i·7-s − 25.8i·9-s + (45.2 − 45.2i)11-s + (−35.3 − 35.3i)13-s + 12.4·15-s − 72.4·17-s + (−19.4 − 19.4i)19-s + (−2.02 + 2.02i)21-s − 139. i·23-s − 10.3i·25-s + (−39.9 + 39.9i)27-s + (−66.0 − 66.0i)29-s + 188.·31-s + ⋯ |
L(s) = 1 | + (−0.145 − 0.145i)3-s + (−0.735 + 0.735i)5-s − 0.144i·7-s − 0.957i·9-s + (1.23 − 1.23i)11-s + (−0.755 − 0.755i)13-s + 0.214·15-s − 1.03·17-s + (−0.234 − 0.234i)19-s + (−0.0210 + 0.0210i)21-s − 1.26i·23-s − 0.0826i·25-s + (−0.285 + 0.285i)27-s + (−0.422 − 0.422i)29-s + 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.591787 - 0.745579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.591787 - 0.745579i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.756 + 0.756i)T + 27iT^{2} \) |
| 5 | \( 1 + (8.22 - 8.22i)T - 125iT^{2} \) |
| 7 | \( 1 + 2.67iT - 343T^{2} \) |
| 11 | \( 1 + (-45.2 + 45.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (35.3 + 35.3i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 72.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + (19.4 + 19.4i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 139. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (66.0 + 66.0i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-84.0 + 84.0i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 104. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-31.4 + 31.4i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (149. - 149. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (284. - 284. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-228. - 228. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (139. + 139. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 453. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 259. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-563. - 563. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 866. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 936.T + 9.12e5T^{2} \) |
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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
−12.37889416131727273861984343988, −11.52487395507013127088692318834, −10.75766173110874836764025995795, −9.358074925010293764187053208309, −8.271070550709058394834791964940, −6.92632332177745948968279783890, −6.16931145807164497436691186714, −4.21837529264336207282542256382, −3.03067288686725272486354180532, −0.50207911117034388433921855295,
1.87551109966510197623493492776, 4.18734888231569326876660036807, 4.90396987855063811890213722988, 6.70938940142311648184905577511, 7.79538792233394554811466649575, 8.991726519391647300236788340767, 9.911307492846467829904286468459, 11.39628167679027497330928173023, 12.00711598750182297918614863936, 13.01091335904261372183436541381