| L(s) = 1 | + 2.82·2-s + 8.00·4-s + 27.8i·5-s + 22.6·8-s + 78.7i·10-s + 139.·11-s − 101. i·13-s + 64.0·16-s + 542. i·17-s + 139. i·19-s + 222. i·20-s + 395.·22-s − 229.·23-s − 150.·25-s − 288. i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 1.11i·5-s + 0.353·8-s + 0.787i·10-s + 1.15·11-s − 0.603i·13-s + 0.250·16-s + 1.87i·17-s + 0.387i·19-s + 0.556i·20-s + 0.816·22-s − 0.434·23-s − 0.240·25-s − 0.426i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.478122735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.478122735\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 27.8iT - 625T^{2} \) |
| 11 | \( 1 - 139.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 101. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 542. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 139. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 229.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 383.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 397. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 898.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 657. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.15e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.29e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.16e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 4.15e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.84e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 6.31e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.26e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.99e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.06e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.75e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.18e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.81e3iT - 8.85e7T^{2} \) |
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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
−10.26455846308005962066810732059, −8.889964313514493176875525473542, −7.966148438453406168987354443989, −6.99481068436637231668432430906, −6.28947206002876355304063726895, −5.64226410698589984551893332966, −4.15608105342759867615968428768, −3.59610640567540339264320358646, −2.50568495334909828582092148002, −1.34566828847757733343170491413,
0.60044295358059914142125294843, 1.64127366422150238210963278696, 2.94076628793862307764540512706, 4.18715838040334664622517300507, 4.76585624823251908055592094334, 5.65933989148956084959664085215, 6.73358322424236619299077928405, 7.41010614309299373990445388142, 8.759278430844290731562387768749, 9.134200794885617482495053237609
