L(s) = 1 | + (0.979 − 1.42i)3-s + (0.5 + 0.866i)5-s + (−0.456 − 2.60i)7-s + (−1.08 − 2.79i)9-s + (0.698 + 0.403i)11-s − 3.86i·13-s + (1.72 + 0.133i)15-s + (−1.05 + 1.83i)17-s + (2.70 − 1.56i)19-s + (−4.17 − 1.89i)21-s + (4.21 − 2.43i)23-s + (−0.499 + 0.866i)25-s + (−5.05 − 1.19i)27-s + 6.67i·29-s + (5.65 + 3.26i)31-s + ⋯ |
L(s) = 1 | + (0.565 − 0.824i)3-s + (0.223 + 0.387i)5-s + (−0.172 − 0.985i)7-s + (−0.360 − 0.932i)9-s + (0.210 + 0.121i)11-s − 1.07i·13-s + (0.445 + 0.0344i)15-s + (−0.257 + 0.445i)17-s + (0.620 − 0.358i)19-s + (−0.910 − 0.414i)21-s + (0.879 − 0.508i)23-s + (−0.0999 + 0.173i)25-s + (−0.973 − 0.229i)27-s + 1.23i·29-s + (1.01 + 0.585i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28347 - 1.02506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28347 - 1.02506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.979 + 1.42i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.456 + 2.60i)T \) |
good | 11 | \( 1 + (-0.698 - 0.403i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.86iT - 13T^{2} \) |
| 17 | \( 1 + (1.05 - 1.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.70 + 1.56i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.21 + 2.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.67iT - 29T^{2} \) |
| 31 | \( 1 + (-5.65 - 3.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.89 + 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 + 0.819T + 43T^{2} \) |
| 47 | \( 1 + (-1.38 - 2.40i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.5 - 6.67i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.86 + 4.95i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.79 + 1.03i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.45 - 9.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.67 - 1.54i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.76 - 11.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.01iT - 97T^{2} \) |
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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
−10.85706735944883731798412842549, −10.20670258835900518473699073819, −9.069641591404454024303865657192, −8.166794002533890842598183231089, −7.13971561172921312548617509523, −6.67761106085128371043298566613, −5.31510949330984381351577988257, −3.72814948360219399748619766156, −2.75513251753761082689450940639, −1.08997491998934139769388866792,
2.08499251404113868543980640584, 3.32108269682448875489956636855, 4.59101891341523113889907124714, 5.44911154489771275392548715267, 6.62478322993740337091047277874, 8.018192296766050149531068384507, 8.896898087523798231578149442304, 9.437112775516869985154349433767, 10.22032831192090655719715919959, 11.60448973234052319728466402759