| L(s) = 1 | + (2.58 + 1.52i)3-s + 2.23i·5-s + 7.48·7-s + (4.32 + 7.89i)9-s + 8.48i·11-s + 10·13-s + (−3.41 + 5.77i)15-s − 30.3i·17-s + 26.9·19-s + (19.3 + 11.4i)21-s − 9.17i·23-s − 5.00·25-s + (−0.905 + 26.9i)27-s + 26.8i·29-s − 8·31-s + ⋯ |
| L(s) = 1 | + (0.860 + 0.509i)3-s + 0.447i·5-s + 1.06·7-s + (0.480 + 0.876i)9-s + 0.771i·11-s + 0.769·13-s + (−0.227 + 0.384i)15-s − 1.78i·17-s + 1.41·19-s + (0.920 + 0.545i)21-s − 0.398i·23-s − 0.200·25-s + (−0.0335 + 0.999i)27-s + 0.925i·29-s − 0.258·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.193588824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.193588824\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.58 - 1.52i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 - 7.48T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 30.3iT - 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + 9.17iT - 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 + 15.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 87.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.0T + 9.40e3T^{2} \) |
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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
−9.817890299212358576805195181268, −9.222867571070003310805321369194, −8.293179518012424931361610958978, −7.52911946122268869905115657791, −6.88670437578468016014561942727, −5.26674870922384965221467466953, −4.73995269754208970589588087759, −3.55033609488002054732154762235, −2.62682297773042309014523442617, −1.43096046073162840379416432940,
1.07785550114039039955309499393, 1.88054806960628135595630255889, 3.35291963688369324165529618072, 4.12418253962500591955009480897, 5.43323522189182917926519060622, 6.23245377410674315847483329622, 7.46446015289677995971139445017, 8.172972237554935193808159951658, 8.599573230549993754729952899126, 9.467735859058012861158558192662
