| L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−1.72 − 2.98i)5-s − 7.99·8-s + (3.44 − 5.96i)10-s + (−18.0 + 31.2i)11-s − 10.2·13-s + (−8 − 13.8i)16-s + (59.2 − 102. i)17-s + (−19.3 − 33.4i)19-s + 13.7·20-s − 72.2·22-s + (18.1 + 31.3i)23-s + (56.5 − 97.9i)25-s + (−10.2 − 17.7i)26-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.154 − 0.266i)5-s − 0.353·8-s + (0.108 − 0.188i)10-s + (−0.495 + 0.857i)11-s − 0.218·13-s + (−0.125 − 0.216i)16-s + (0.845 − 1.46i)17-s + (−0.233 − 0.404i)19-s + 0.154·20-s − 0.700·22-s + (0.164 + 0.284i)23-s + (0.452 − 0.783i)25-s + (−0.0771 − 0.133i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.995679349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995679349\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (1.72 + 2.98i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (18.0 - 31.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 10.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-59.2 + 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (19.3 + 33.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-18.1 - 31.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 12.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + (72.7 - 126. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-0.685 - 1.18i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-251. - 435. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-312. + 541. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (21.1 - 36.5i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (219. + 380. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-381. + 661. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-289. + 501. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (471. + 816. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 474.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-410. - 711. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−9.607276177377173901830631705675, −8.906673472997669470122469290486, −7.80369194273936652584994815100, −7.29772078315525357896952144248, −6.36843824990308214691851340835, −5.12137685166680639139669685420, −4.77515735588400412536330839092, −3.45752430245226131212810379773, −2.33830701193487504980217854795, −0.59071862057341245600669537812,
0.910833271251803366621990951643, 2.23753613263339476819979896610, 3.35284074710750304906951083001, 4.08034593064765230631048702574, 5.41773293776203551519467865543, 5.97516994416949337470229376732, 7.18720413458699469275585127708, 8.160655216358189898002243567673, 8.906029348120300134624117117957, 10.02306698341281838022612667804
