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
−8.321184264662958831380184924048, −7.57718686881061823115386782069, −6.82276817270902074991973951434, −6.08798915012106233594360015949, −5.23199878991683042265696078592, −4.77073260213750673628241997163, −3.64545691194168901397386212390, −2.91304710361140071649302092555, −1.63892642457084189369074284504, −0.45603672213089351982205013068,
0.945907123114832266594669379893, 1.97745748178890557633293797471, 3.29605277568759525785473926079, 4.11476411781896846137870525743, 4.72987511414251221478862092179, 5.62926903962508381435732929118, 6.41835282355829746781294154539, 7.06176482320977662297657931542, 7.84026499894182873168525969254, 8.412140352764434818279958631877