L(s) = 1 | − 1.41·5-s + 8·7-s + 11.3·11-s − 8i·13-s + 12.7i·17-s − 32i·19-s − 33.9i·23-s − 23·25-s + 43.8·29-s − 40·31-s − 11.3·35-s + 26i·37-s − 66.4i·41-s + 16i·43-s − 11.3i·47-s + ⋯ |
L(s) = 1 | − 0.282·5-s + 1.14·7-s + 1.02·11-s − 0.615i·13-s + 0.748i·17-s − 1.68i·19-s − 1.47i·23-s − 0.920·25-s + 1.51·29-s − 1.29·31-s − 0.323·35-s + 0.702i·37-s − 1.62i·41-s + 0.372i·43-s − 0.240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.069119050\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069119050\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41T + 25T^{2} \) |
| 7 | \( 1 - 8T + 49T^{2} \) |
| 11 | \( 1 - 11.3T + 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 - 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 32iT - 361T^{2} \) |
| 23 | \( 1 + 33.9iT - 529T^{2} \) |
| 29 | \( 1 - 43.8T + 841T^{2} \) |
| 31 | \( 1 + 40T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 66.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 22.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 54iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 80iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 - 104T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80T + 9.40e3T^{2} \) |
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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
−8.551569148911007970074900405631, −8.055746530747326860438290561725, −7.07754222858319564937970894983, −6.43600645853625007180072825804, −5.38736126733883013672116131371, −4.59114699715124120187282819571, −3.92405953509804139886141578797, −2.72369529685937526103488588961, −1.65981445682039772320372174541, −0.52731197680205007433539628504,
1.25945247615183311690766855539, 1.92919329826591436508112531798, 3.42829475589692927911913334673, 4.13201178127763089585665144340, 4.97470192043978893374893742831, 5.84999607375970083617114219141, 6.69409034566485266241086782579, 7.71588489447979623394327014560, 7.990776892736681279825906435449, 9.096571244655334849589673830858