| 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
−11.81253560056652534402273889487, −11.23351291842162436193660909209, −10.29352711585917684055578521472, −8.981481467728441884631512473507, −8.090793288220143307987613609554, −7.24918847533424344565727009936, −4.98191280746279973658128206760, −4.08547432877963133608036257576, −3.00851032816144077645042275398, −0.45299138851756888606780668230,
3.08199285899863100861061968640, 4.78951215672886771984548155068, 5.41100638383509963649112167785, 7.44232142157706463302941399565, 7.76850104191681255845394607995, 8.434216158356857950883117159359, 9.925083547951419607168120032263, 11.13076184727280563181787857748, 12.19726343676626929966945769778, 12.92435397842830025236247464543
