L(s) = 1 | + (−395. + 606. i)2-s + (−2.71e4 − 2.06e4i)3-s + (−2.11e5 − 4.79e5i)4-s − 7.30e5i·5-s + (2.32e7 − 8.29e6i)6-s + 7.02e6i·7-s + (3.74e8 + 6.12e7i)8-s + (3.09e8 + 1.12e9i)9-s + (4.43e8 + 2.88e8i)10-s + 9.08e9·11-s + (−4.15e9 + 1.73e10i)12-s − 2.88e10·13-s + (−4.26e9 − 2.77e9i)14-s + (−1.50e10 + 1.98e10i)15-s + (−1.85e11 + 2.03e11i)16-s + 3.88e11i·17-s + ⋯ |
L(s) = 1 | + (−0.546 + 0.837i)2-s + (−0.795 − 0.605i)3-s + (−0.403 − 0.914i)4-s − 0.167i·5-s + (0.941 − 0.336i)6-s + 0.0658i·7-s + (0.986 + 0.161i)8-s + (0.266 + 0.963i)9-s + (0.140 + 0.0913i)10-s + 1.16·11-s + (−0.232 + 0.972i)12-s − 0.755·13-s + (−0.0551 − 0.0359i)14-s + (−0.101 + 0.133i)15-s + (−0.673 + 0.738i)16-s + 0.794i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(0.5151243510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5151243510\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (395. - 606. i)T \) |
| 3 | \( 1 + (2.71e4 + 2.06e4i)T \) |
good | 5 | \( 1 + 7.30e5iT - 1.90e13T^{2} \) |
| 7 | \( 1 - 7.02e6iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 9.08e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.88e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 3.88e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 + 1.03e12iT - 1.97e24T^{2} \) |
| 23 | \( 1 - 2.94e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 8.24e13iT - 6.10e27T^{2} \) |
| 31 | \( 1 - 2.06e12iT - 2.16e28T^{2} \) |
| 37 | \( 1 + 1.42e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.17e15iT - 4.39e30T^{2} \) |
| 43 | \( 1 + 3.15e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 + 8.86e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.81e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 + 1.21e17T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.00e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.94e17iT - 4.95e34T^{2} \) |
| 71 | \( 1 + 5.81e16T + 1.49e35T^{2} \) |
| 73 | \( 1 + 3.90e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.66e18iT - 1.13e36T^{2} \) |
| 83 | \( 1 - 1.76e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 1.08e18iT - 1.09e37T^{2} \) |
| 97 | \( 1 - 8.25e18T + 5.60e37T^{2} \) |
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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
−15.29085782365172617351669580388, −13.80862685257145522667867118942, −12.20685265991111872881744603688, −10.62259386551977676966666248906, −8.954342401244154937914102156405, −7.32033055986811253671029871104, −6.20943451020240394180697708529, −4.74982046692522468200896381738, −1.60441105577807598169596261292, −0.25629224598614410909920647828,
1.25080057586313120908499009284, 3.33081347854913436143987992782, 4.80072248003983213466976828912, 6.93465328593272969046210493916, 9.050476635341304302257136520060, 10.19618688943886888057380028176, 11.45937459266538358285103151921, 12.45236472094745806071237051724, 14.46499919491340902358926020224, 16.35219371358912854573135941474