| L(s) = 1 | + (0.996 − 1.00i)2-s + (−0.463 + 1.72i)3-s + (−0.0138 − 1.99i)4-s + (−1.91 + 1.16i)5-s + (1.27 + 2.18i)6-s + (−2.02 − 1.97i)8-s + (−0.177 − 0.102i)9-s + (−0.737 + 3.07i)10-s + (−1.83 + 1.05i)11-s + (3.46 + 0.902i)12-s + (2.86 − 2.86i)13-s + (−1.12 − 3.84i)15-s + (−3.99 + 0.0552i)16-s + (−1.63 + 6.10i)17-s + (−0.279 + 0.0759i)18-s + (−1.47 + 2.55i)19-s + ⋯ |
| L(s) = 1 | + (0.704 − 0.709i)2-s + (−0.267 + 0.998i)3-s + (−0.00690 − 0.999i)4-s + (−0.854 + 0.519i)5-s + (0.519 + 0.893i)6-s + (−0.714 − 0.699i)8-s + (−0.0591 − 0.0341i)9-s + (−0.233 + 0.972i)10-s + (−0.551 + 0.318i)11-s + (1.00 + 0.260i)12-s + (0.794 − 0.794i)13-s + (−0.290 − 0.991i)15-s + (−0.999 + 0.0138i)16-s + (−0.396 + 1.48i)17-s + (−0.0659 + 0.0179i)18-s + (−0.338 + 0.586i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472831 + 0.759128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472831 + 0.759128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 + 1.00i)T \) |
| 5 | \( 1 + (1.91 - 1.16i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (0.463 - 1.72i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.86 + 2.86i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.63 - 6.10i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.47 - 2.55i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.38 - 1.44i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.86iT - 29T^{2} \) |
| 31 | \( 1 + (5.20 - 3.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.34 - 1.43i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + (-5.64 - 5.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.57 + 5.87i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.766 - 0.205i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.48 + 4.31i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.843 - 1.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.0 - 2.95i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 0.610iT - 71T^{2} \) |
| 73 | \( 1 + (-4.41 - 1.18i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.12 + 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.51 - 5.51i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.80 + 1.03i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.95 + 1.95i)T + 97iT^{2} \) |
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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
−10.57390071498383086172958773392, −9.963727996602829773511199385936, −8.713593635940661368028752567612, −7.84960990316500227389011551872, −6.61292713146942553366882356735, −5.68894427731757430978211683681, −4.81289975302523589220517271717, −3.78822629358935024451880753272, −3.49754657014011682566422055173, −1.87782758063649494043580996411,
0.32363929968578869853675195465, 2.19457176754424139694126257994, 3.64263727940249141588692064041, 4.49271359119162021791167886144, 5.46404482956994786524380556211, 6.46015673807938988462523032314, 7.12310380803680949429076318367, 7.79961639451566090625424615119, 8.575322554066672740025002389178, 9.369677069268696192279658095073
