| L(s) = 1 | + i·3-s + (2 − 2i)5-s + (−2 + 2i)7-s − 9-s + (−3 + 3i)11-s + (−3 − 2i)13-s + (2 + 2i)15-s + 2i·17-s + (−2 − 2i)21-s − 6·23-s − 3i·25-s − i·27-s − 6·29-s + (−3 − 3i)33-s + 8i·35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.894i)5-s + (−0.755 + 0.755i)7-s − 0.333·9-s + (−0.904 + 0.904i)11-s + (−0.832 − 0.554i)13-s + (0.516 + 0.516i)15-s + 0.485i·17-s + (−0.436 − 0.436i)21-s − 1.25·23-s − 0.600i·25-s − 0.192i·27-s − 1.11·29-s + (−0.522 − 0.522i)33-s + 1.35i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5379782094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5379782094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + (-2 + 2i)T - 5iT^{2} \) |
| 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31iT^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (8 - 8i)T - 41iT^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 5i)T - 47iT^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-1 + i)T - 59iT^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-2 - 2i)T + 67iT^{2} \) |
| 71 | \( 1 + (5 + 5i)T + 71iT^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (-5 - 5i)T + 83iT^{2} \) |
| 89 | \( 1 + (10 + 10i)T + 89iT^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−9.819415813263351810294594741103, −9.568510473537187318260571456556, −8.574713639651475020678958520628, −7.81745793678103560859780626306, −6.59787089566926869138793587261, −5.52272083857501347831334809524, −5.27342706974055570665087963068, −4.15250512055590314613902013654, −2.81406375762341700323020629736, −1.92117587137868663663008959122,
0.20130663919602476001420235636, 2.05662660746237516437146587996, 2.85700844573537960160768581338, 3.90086267490400128523161431674, 5.39991640172074913203007792586, 6.06073918143313924680503430491, 6.95210555341148299801362840287, 7.39748194751257318251546694496, 8.461837929987037318152368865274, 9.564548255447177082610828469577
