| L(s) = 1 | + (−0.0502 + 1.41i)2-s + (0.268 − 0.182i)3-s + (−1.99 − 0.142i)4-s + (−3.87 + 0.290i)5-s + (0.244 + 0.388i)6-s + (1.56 + 2.13i)7-s + (0.301 − 2.81i)8-s + (−1.05 + 2.69i)9-s + (−0.215 − 5.49i)10-s + (−3.83 + 1.50i)11-s + (−0.561 + 0.326i)12-s + (−3.48 − 2.77i)13-s + (−3.09 + 2.10i)14-s + (−0.986 + 0.786i)15-s + (3.95 + 0.567i)16-s + (1.72 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.0355 + 0.999i)2-s + (0.154 − 0.105i)3-s + (−0.997 − 0.0710i)4-s + (−1.73 + 0.129i)5-s + (0.100 + 0.158i)6-s + (0.590 + 0.807i)7-s + (0.106 − 0.994i)8-s + (−0.352 + 0.898i)9-s + (−0.0681 − 1.73i)10-s + (−1.15 + 0.454i)11-s + (−0.161 + 0.0943i)12-s + (−0.965 − 0.770i)13-s + (−0.827 + 0.561i)14-s + (−0.254 + 0.203i)15-s + (0.989 + 0.141i)16-s + (0.419 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0235105 - 0.514084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0235105 - 0.514084i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0502 - 1.41i)T \) |
| 7 | \( 1 + (-1.56 - 2.13i)T \) |
| good | 3 | \( 1 + (-0.268 + 0.182i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (3.87 - 0.290i)T + (4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (3.83 - 1.50i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (3.48 + 2.77i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 1.86i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.67 - 4.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 2.10i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.966 - 4.23i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.62 + 2.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.52 - 0.779i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.642i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.62 - 7.52i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (5.43 + 0.819i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (1.19 + 0.368i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.170 + 2.28i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (2.41 + 7.84i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-9.53 - 5.50i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.3 - 2.36i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.802 - 5.32i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-0.555 + 0.320i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.40 + 4.27i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (3.68 + 1.44i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 0.875iT - 97T^{2} \) |
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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
−12.92435397842830025236247464543, −12.19726343676626929966945769778, −11.13076184727280563181787857748, −9.925083547951419607168120032263, −8.434216158356857950883117159359, −7.76850104191681255845394607995, −7.44232142157706463302941399565, −5.41100638383509963649112167785, −4.78951215672886771984548155068, −3.08199285899863100861061968640,
0.45299138851756888606780668230, 3.00851032816144077645042275398, 4.08547432877963133608036257576, 4.98191280746279973658128206760, 7.24918847533424344565727009936, 8.090793288220143307987613609554, 8.981481467728441884631512473507, 10.29352711585917684055578521472, 11.23351291842162436193660909209, 11.81253560056652534402273889487
