L(s) = 1 | + 1.73i·3-s + 3.46i·5-s + 3.46i·7-s − 2.99·9-s + 6·11-s − 13-s − 5.99·15-s − 6.92i·17-s + 3.46i·19-s − 5.99·21-s − 6.99·25-s − 5.19i·27-s − 6.92i·29-s + 3.46i·31-s + 10.3i·33-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 1.54i·5-s + 1.30i·7-s − 0.999·9-s + 1.80·11-s − 0.277·13-s − 1.54·15-s − 1.68i·17-s + 0.794i·19-s − 1.30·21-s − 1.39·25-s − 0.999i·27-s − 1.28i·29-s + 0.622i·31-s + 1.80i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.376811 + 1.40628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.376811 + 1.40628i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−11.02800999141591320291254457519, −9.940868885026710128992222313955, −9.433390906532394122006901798006, −8.633706583848249850925095462068, −7.31226406221609535357419785777, −6.35126912991818833584255352855, −5.64719839344910413032495237364, −4.32123708202102783104488821035, −3.24734251158774712908038129070, −2.42821001225141302353614008488,
0.873744568762167675593204558476, 1.63695037687821248081465144622, 3.74727628424964111313391801131, 4.53303760030667864849983903569, 5.82181096687217762589216170136, 6.75894715689623763017263684003, 7.54439740048871930833255861244, 8.637417657495881218126270841227, 9.009000933475639275001537530407, 10.23276379574503269036406319956