L(s) = 1 | + (0.930 + 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.432 − 0.294i)5-s + (−1.64 − 2.07i)7-s + (0.433 + 0.900i)8-s + (−0.294 − 0.432i)10-s + (−0.320 − 2.12i)11-s + (2.39 − 1.90i)13-s + (−0.774 − 2.52i)14-s + (0.0747 + 0.997i)16-s + (4.61 + 1.42i)17-s + (4.76 − 2.75i)19-s + (−0.116 − 0.510i)20-s + (0.478 − 2.09i)22-s + (−0.0351 − 0.113i)23-s + ⋯ |
L(s) = 1 | + (0.658 + 0.258i)2-s + (0.366 + 0.340i)4-s + (−0.193 − 0.131i)5-s + (−0.621 − 0.783i)7-s + (0.153 + 0.318i)8-s + (−0.0932 − 0.136i)10-s + (−0.0966 − 0.640i)11-s + (0.663 − 0.529i)13-s + (−0.207 − 0.676i)14-s + (0.0186 + 0.249i)16-s + (1.11 + 0.345i)17-s + (1.09 − 0.631i)19-s + (−0.0260 − 0.114i)20-s + (0.101 − 0.446i)22-s + (−0.00732 − 0.0237i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07084 - 0.564271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07084 - 0.564271i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.930 - 0.365i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.64 + 2.07i)T \) |
good | 5 | \( 1 + (0.432 + 0.294i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.320 + 2.12i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-2.39 + 1.90i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.61 - 1.42i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-4.76 + 2.75i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0351 + 0.113i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.76 + 1.08i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.23 + 0.714i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.24 + 4.86i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.904 - 0.435i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (3.78 + 1.82i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.33 - 11.0i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.364 + 0.392i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (9.74 - 6.64i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (0.182 + 0.196i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.72 + 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.17 - 0.496i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (7.27 - 2.85i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (4.67 + 8.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.0574 + 0.0720i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 1.61i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−10.19508225584496121269047160990, −9.245306891975243585847483379428, −8.065259101918527450006273373776, −7.57668953174006181399620322016, −6.42253390432116090758144020541, −5.80264011567818300092958955509, −4.68770753070240052275152725028, −3.63704617231926909680463199539, −2.95792846535782828841759757929, −0.925448360557786541199399404021,
1.55292867178210050044930800549, 2.96023146788011493403098111304, 3.66752030268005825978305495877, 4.96382803907347644074731196902, 5.73123828682248774729910875549, 6.63645897644507227098866948476, 7.52192154786968455108881476865, 8.558089035820596727397304342501, 9.690421684895424115198206952648, 10.02153446094295337215589117578