L(s) = 1 | − 3-s − 5-s + (0.637 − 1.10i)7-s + 9-s + (−0.198 + 0.343i)11-s + (1.57 + 2.73i)13-s + 15-s + (−0.405 − 0.701i)17-s + (2.24 + 3.89i)19-s + (−0.637 + 1.10i)21-s + (−0.399 − 0.691i)23-s + 25-s − 27-s + (2.44 − 4.23i)29-s + (−0.134 + 0.233i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + (0.241 − 0.417i)7-s + 0.333·9-s + (−0.0597 + 0.103i)11-s + (0.437 + 0.757i)13-s + 0.258·15-s + (−0.0982 − 0.170i)17-s + (0.515 + 0.892i)19-s + (−0.139 + 0.241i)21-s + (−0.0832 − 0.144i)23-s + 0.200·25-s − 0.192·27-s + (0.454 − 0.787i)29-s + (−0.0242 + 0.0419i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.160878644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.160878644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (0.366 - 8.17i)T \) |
good | 7 | \( 1 + (-0.637 + 1.10i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.198 - 0.343i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 2.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.405 + 0.701i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.24 - 3.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.399 + 0.691i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.44 + 4.23i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.134 - 0.233i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.34 + 4.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.47 - 6.01i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + (-1.85 + 3.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + (1.93 + 3.34i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-2.11 + 3.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.97 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.255 - 0.442i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.92 - 3.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.151T + 89T^{2} \) |
| 97 | \( 1 + (-3.70 - 6.41i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.412140352764434818279958631877, −7.84026499894182873168525969254, −7.06176482320977662297657931542, −6.41835282355829746781294154539, −5.62926903962508381435732929118, −4.72987511414251221478862092179, −4.11476411781896846137870525743, −3.29605277568759525785473926079, −1.97745748178890557633293797471, −0.945907123114832266594669379893,
0.45603672213089351982205013068, 1.63892642457084189369074284504, 2.91304710361140071649302092555, 3.64545691194168901397386212390, 4.77073260213750673628241997163, 5.23199878991683042265696078592, 6.08798915012106233594360015949, 6.82276817270902074991973951434, 7.57718686881061823115386782069, 8.321184264662958831380184924048