| 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
−10.02306698341281838022612667804, −8.906029348120300134624117117957, −8.160655216358189898002243567673, −7.18720413458699469275585127708, −5.97516994416949337470229376732, −5.41773293776203551519467865543, −4.08034593064765230631048702574, −3.35284074710750304906951083001, −2.23753613263339476819979896610, −0.910833271251803366621990951643,
0.59071862057341245600669537812, 2.33830701193487504980217854795, 3.45752430245226131212810379773, 4.77515735588400412536330839092, 5.12137685166680639139669685420, 6.36843824990308214691851340835, 7.29772078315525357896952144248, 7.80369194273936652584994815100, 8.906673472997669470122469290486, 9.607276177377173901830631705675
