L(s) = 1 | + 2.23i·5-s − 5.48·7-s − 9.17i·11-s + 11.4·13-s + 16.9i·17-s − 26.9·19-s − 4.93i·23-s − 5.00·25-s − 20.5i·29-s − 20.9·31-s − 12.2i·35-s − 62.4·37-s − 40.9i·41-s − 1.02·43-s − 86.2i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s − 0.783·7-s − 0.833i·11-s + 0.883·13-s + 0.998i·17-s − 1.41·19-s − 0.214i·23-s − 0.200·25-s − 0.707i·29-s − 0.676·31-s − 0.350i·35-s − 1.68·37-s − 0.998i·41-s − 0.0238·43-s − 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3727656734\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3727656734\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 + 5.48T + 49T^{2} \) |
| 11 | \( 1 + 9.17iT - 121T^{2} \) |
| 13 | \( 1 - 11.4T + 169T^{2} \) |
| 17 | \( 1 - 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 + 4.93iT - 529T^{2} \) |
| 29 | \( 1 + 20.5iT - 841T^{2} \) |
| 31 | \( 1 + 20.9T + 961T^{2} \) |
| 37 | \( 1 + 62.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.02T + 1.84e3T^{2} \) |
| 47 | \( 1 + 86.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 96.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76T + 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 45.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.0T + 9.40e3T^{2} \) |
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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
−10.09544619789351730446619455476, −8.758485041604050509143428486594, −8.426055633528524670749612431259, −7.04476849637066490497916188646, −6.31133656697192482564604904614, −5.61352667230731727304522876065, −4.01130569334386354094803118491, −3.34922981298840503628692610327, −1.95816018837077568326704633334, −0.12597981920862756534044671369,
1.57156501231235086105516507672, 2.99358628069943407213453880940, 4.13436518058791019141432267009, 5.09093116128727799445949974509, 6.22899436201162062239096346755, 6.95934374592816069682582675446, 8.008797291100243424331887022152, 9.005134531376907263508863144840, 9.545791488901155140496902840964, 10.54673742115730349176436038241