| L(s) = 1 | + (−0.930 + 0.365i)2-s + (1.16 − 1.27i)3-s + (0.733 − 0.680i)4-s + (3.54 − 2.41i)5-s + (−0.619 + 1.61i)6-s + (0.973 + 2.46i)7-s + (−0.433 + 0.900i)8-s + (−0.274 − 2.98i)9-s + (−2.41 + 3.54i)10-s + (−0.317 + 2.10i)11-s + (−0.0145 − 1.73i)12-s + (0.621 + 0.495i)13-s + (−1.80 − 1.93i)14-s + (1.04 − 7.36i)15-s + (0.0747 − 0.997i)16-s + (−7.16 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 + 0.258i)2-s + (0.673 − 0.738i)3-s + (0.366 − 0.340i)4-s + (1.58 − 1.08i)5-s + (−0.252 + 0.660i)6-s + (0.367 + 0.929i)7-s + (−0.153 + 0.318i)8-s + (−0.0914 − 0.995i)9-s + (−0.765 + 1.12i)10-s + (−0.0957 + 0.635i)11-s + (−0.00420 − 0.499i)12-s + (0.172 + 0.137i)13-s + (−0.482 − 0.516i)14-s + (0.270 − 1.90i)15-s + (0.0186 − 0.249i)16-s + (−1.73 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40634 - 0.464275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40634 - 0.464275i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.16 + 1.27i)T \) |
| 7 | \( 1 + (-0.973 - 2.46i)T \) |
| good | 5 | \( 1 + (-3.54 + 2.41i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (0.317 - 2.10i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.621 - 0.495i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (7.16 - 2.21i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (1.88 + 1.08i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.36 - 4.43i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-6.59 - 1.50i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.56 - 1.48i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.202 - 0.187i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (5.29 + 2.54i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.05 + 1.95i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.177 - 0.451i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (3.48 + 3.75i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (3.31 + 2.26i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (1.09 - 1.18i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.16 - 3.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.36 - 0.540i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (11.5 + 4.52i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-5.04 + 8.74i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.49 - 9.39i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.32 + 0.803i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + 3.52iT - 97T^{2} \) |
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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
−11.87881387464725554613046192295, −10.43977014802321106803375220822, −9.277645556045365634853632368511, −8.950256022323884352356592573159, −8.194637591202660129393403660539, −6.72568843316875315989769972934, −5.96430201535233678258619535133, −4.80466382467439412508456889393, −2.30270609053053383247974156453, −1.66149007037297824206324079622,
2.06706684860733592834350772301, 3.04375963305569213371545345025, 4.53818737741043830303533127266, 6.13593890401865971227387019606, 7.04975820098991367910196917583, 8.326709011378899488246353198072, 9.215488363048619492217652102294, 10.11458304291874815538006317294, 10.67866918112596620955589717043, 11.18503654805816045423064166226
