| L(s) = 1 | + (−1 + 2.82i)3-s + 4.24i·5-s + (−7.00 − 5.65i)9-s − 4.24i·11-s + 4·13-s + (−12 − 4.24i)15-s − 7.07i·17-s − 16·19-s − 41.0i·23-s + 7.00·25-s + (23.0 − 14.1i)27-s − 22.6i·29-s + 50·31-s + (12 + 4.24i)33-s − 48·37-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)3-s + 0.848i·5-s + (−0.777 − 0.628i)9-s − 0.385i·11-s + 0.307·13-s + (−0.800 − 0.282i)15-s − 0.415i·17-s − 0.842·19-s − 1.78i·23-s + 0.280·25-s + (0.851 − 0.523i)27-s − 0.780i·29-s + 1.61·31-s + (0.363 + 0.128i)33-s − 1.29·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.182541860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182541860\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 2.82i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.24iT - 25T^{2} \) |
| 11 | \( 1 + 4.24iT - 121T^{2} \) |
| 13 | \( 1 - 4T + 169T^{2} \) |
| 17 | \( 1 + 7.07iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 41.0iT - 529T^{2} \) |
| 29 | \( 1 + 22.6iT - 841T^{2} \) |
| 31 | \( 1 - 50T + 961T^{2} \) |
| 37 | \( 1 + 48T + 1.36e3T^{2} \) |
| 41 | \( 1 + 4.24iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 44T + 1.84e3T^{2} \) |
| 47 | \( 1 + 28.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 68T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44T + 4.48e3T^{2} \) |
| 71 | \( 1 - 4.24iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 128T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80T + 6.24e3T^{2} \) |
| 83 | \( 1 + 124. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 162. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 136T + 9.40e3T^{2} \) |
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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
−9.712271547077739398771432138533, −8.724893449991324487600216233599, −8.132030629716734375671379579312, −6.61075175011483132789526379489, −6.41446157235534335023789652529, −5.16573009264613671306072989965, −4.33556871098246385030727730048, −3.34186007366747418234670292766, −2.46099373722823308133038957830, −0.42268238411988768498418881591,
1.07402070322874782641427421625, 1.93295019619988039395714566856, 3.34818421866223530722534687458, 4.66872931272927841522819875840, 5.40059988078797688989228435688, 6.34404479718877071207397282175, 7.09144537132476413391974882842, 8.082818611401989478230408477876, 8.590703926255990635121080605562, 9.512778772263167093442140109023
