| L(s) = 1 | + (−2.27 − 1.01i)2-s + (0.697 + 0.148i)3-s + (2.81 + 3.12i)4-s + (−0.230 + 2.19i)5-s + (−1.43 − 1.04i)6-s + (−2.62 + 0.304i)7-s + (−1.69 − 5.22i)8-s + (−2.27 − 1.01i)9-s + (2.74 − 4.75i)10-s + (1.49 + 2.59i)12-s + (2.65 − 1.93i)13-s + (6.28 + 1.96i)14-s + (−0.486 + 1.49i)15-s + (−0.550 + 5.23i)16-s + (1.36 − 0.606i)17-s + (4.15 + 4.61i)18-s + ⋯ |
| L(s) = 1 | + (−1.60 − 0.716i)2-s + (0.402 + 0.0856i)3-s + (1.40 + 1.56i)4-s + (−0.103 + 0.980i)5-s + (−0.587 − 0.426i)6-s + (−0.993 + 0.115i)7-s + (−0.599 − 1.84i)8-s + (−0.758 − 0.337i)9-s + (0.868 − 1.50i)10-s + (0.433 + 0.750i)12-s + (0.737 − 0.535i)13-s + (1.68 + 0.526i)14-s + (−0.125 + 0.386i)15-s + (−0.137 + 1.30i)16-s + (0.330 − 0.147i)17-s + (0.978 + 1.08i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.129562 - 0.267151i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.129562 - 0.267151i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.62 - 0.304i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.27 + 1.01i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.697 - 0.148i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (0.230 - 2.19i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-2.65 + 1.93i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.36 + 0.606i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.62 - 5.14i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (3.24 + 5.62i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.509 - 1.56i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.245 - 2.33i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-5.43 + 1.15i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (3.47 + 10.6i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + (-0.997 + 1.10i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.0318 + 0.303i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (8.47 + 9.40i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 12.9i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.28 + 3.96i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.16 + 6.65i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.73 + 6.36i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-4.23 - 1.88i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.56 - 1.13i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.60 + 2.77i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.50 - 1.09i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.953804914944081466018787785823, −9.075780892353967834689323843113, −8.438797148347051568516317217380, −7.70896275070763127517500631968, −6.62374552950292522115889124964, −5.99593436641222621517370871580, −3.67457830877614403770554296262, −3.08637479298780401168257344061, −2.15603101256170167449041963030, −0.25190968488832074877827373275,
1.19622085081262293567362113067, 2.64197243103177416070701438033, 4.22032446823634186262976053927, 5.71582201998331466864804147269, 6.33996363260153433026788989812, 7.37112869768913680046540027208, 8.169654999267447361865347351855, 8.863240708509965802469752520672, 9.252141183301195964279839317731, 10.08947670942862087696094470726
