| L(s) = 1 | + (−1.41 + 0.0602i)2-s + (1.01 − 0.253i)3-s + (1.99 − 0.170i)4-s + (−2.83 − 1.34i)5-s + (−1.41 + 0.419i)6-s + (−0.632 − 2.08i)7-s + (−2.80 + 0.360i)8-s + (−1.68 + 0.900i)9-s + (4.09 + 1.72i)10-s + (−4.58 + 0.679i)11-s + (1.97 − 0.678i)12-s + (−0.0701 + 0.196i)13-s + (1.01 + 2.90i)14-s + (−3.21 − 0.639i)15-s + (3.94 − 0.678i)16-s + (−1.90 + 0.378i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0425i)2-s + (0.584 − 0.146i)3-s + (0.996 − 0.0850i)4-s + (−1.26 − 0.600i)5-s + (−0.578 + 0.171i)6-s + (−0.238 − 0.787i)7-s + (−0.991 + 0.127i)8-s + (−0.561 + 0.300i)9-s + (1.29 + 0.545i)10-s + (−1.38 + 0.204i)11-s + (0.570 − 0.195i)12-s + (−0.0194 + 0.0544i)13-s + (0.272 + 0.776i)14-s + (−0.830 − 0.165i)15-s + (0.985 − 0.169i)16-s + (−0.461 + 0.0917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0509431 - 0.275839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0509431 - 0.275839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0602i)T \) |
| good | 3 | \( 1 + (-1.01 + 0.253i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (2.83 + 1.34i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.632 + 2.08i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (4.58 - 0.679i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.0701 - 0.196i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (1.90 - 0.378i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (0.0488 + 0.0443i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-7.00 + 0.690i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (3.55 + 4.79i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (9.08 + 3.76i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.490 + 9.98i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-7.56 + 9.22i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (2.00 - 7.98i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-5.11 - 3.41i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (1.75 - 2.36i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-0.442 - 1.23i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-3.26 + 5.43i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (0.0461 + 0.0276i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (1.19 - 2.23i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (2.24 - 7.41i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.27 - 1.91i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.250 + 5.09i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-0.0663 - 0.00653i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (0.00361 - 0.00873i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.12768968216050117842297617839, −10.87775183273021468742541117807, −9.402216777308407503772808214902, −8.604608236471861385968962222247, −7.59940256857443009079167934720, −7.39370840662420104013092166763, −5.50468850182976045842966479681, −3.91200811335468296500709589059, −2.50908644919776401399702504293, −0.26002227663469607645794926964,
2.68917501888760535511473795906, 3.36705108097778350803588241891, 5.45909557699580222044691795451, 6.89968909148679955647387778064, 7.77168803066368474773915338846, 8.597772333963012794907858597811, 9.288077652017051655419628380691, 10.66182481962301824231482765794, 11.22881449100972987281079008706, 12.13337392755455355358414564923
