L(s) = 1 | − 3.25i·3-s + 2.23·5-s − 2.64i·7-s − 7.62·9-s − 0.359i·11-s − 5.64·13-s − 7.28i·15-s + 7.99·17-s − 8.62·21-s + 5.00·25-s + 15.0i·27-s − 0.623·29-s − 1.17·33-s − 5.91i·35-s + 18.4i·39-s + ⋯ |
L(s) = 1 | − 1.88i·3-s + 0.999·5-s − 0.999i·7-s − 2.54·9-s − 0.108i·11-s − 1.56·13-s − 1.88i·15-s + 1.93·17-s − 1.88·21-s + 1.00·25-s + 2.90i·27-s − 0.115·29-s − 0.204·33-s − 0.999i·35-s + 2.94i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.386476 - 1.44234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.386476 - 1.44234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 3.25iT - 3T^{2} \) |
| 11 | \( 1 + 0.359iT - 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 - 7.99T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 0.623T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 8.00iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−10.31605383967082304042367990946, −9.611312957904057398561433676300, −8.314220718866060190677614120693, −7.45411928856384357447737804322, −7.00000736300144612746603081610, −5.96644578573700529650710068072, −5.13524599227623388318201682493, −3.13638647498703291988239485388, −2.00604304111960280139161345393, −0.865849303100273279336043796972,
2.44320713941922778342131863683, 3.35919897074833712652306823351, 4.85997141805104337347500768284, 5.30947677007664901562632792330, 6.12975616063564996868066168859, 7.80709462179260308760873969039, 8.935413303568227530171058822969, 9.660316736400341679149232324204, 9.927686904027745764127690245511, 10.81670730510307903924276566983