L(s) = 1 | + (9.84e3 + 5.68e3i)3-s + (−6.55e4 + 1.13e5i)4-s + (−1.37e6 − 1.51e7i)7-s + (6.45e7 + 1.11e8i)9-s + (−1.28e9 + 7.44e8i)12-s + 4.89e9i·13-s + (−8.58e9 − 1.48e10i)16-s + (−1.26e11 + 7.33e10i)19-s + (7.27e10 − 1.57e11i)21-s + (3.81e11 − 6.60e11i)25-s + 1.46e12i·27-s + (1.81e12 + 8.38e11i)28-s + (−4.37e12 − 2.52e12i)31-s − 1.69e13·36-s + (−2.10e13 − 3.64e13i)37-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (−0.0904 − 0.995i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s + 1.66i·13-s + (−0.499 − 0.866i)16-s + (−1.71 + 0.990i)19-s + (0.419 − 0.907i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.907 + 0.419i)28-s + (−0.920 − 0.531i)31-s − 0.999·36-s + (−0.985 − 1.70i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(0.8600789367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8600789367\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-9.84e3 - 5.68e3i)T \) |
| 7 | \( 1 + (1.37e6 + 1.51e7i)T \) |
good | 2 | \( 1 + (6.55e4 - 1.13e5i)T^{2} \) |
| 5 | \( 1 + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 - 4.89e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (1.26e11 - 7.33e10i)T + (2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 - 7.25e24T^{2} \) |
| 31 | \( 1 + (4.37e12 + 2.52e12i)T + (1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (2.10e13 + 3.64e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.48e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (2.59e15 - 1.49e15i)T + (1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-1.34e14 + 2.33e14i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 - 2.96e31T^{2} \) |
| 73 | \( 1 + (-7.49e15 - 4.32e15i)T + (2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (-1.32e16 - 2.29e16i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 + 4.21e32T^{2} \) |
| 89 | \( 1 + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 + 9.91e16iT - 5.95e33T^{2} \) |
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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
−14.52442582796849976908594628050, −13.74031202303732162307301646737, −12.52906423313397648282384532445, −10.72987156030919159941267717933, −9.334461195804108735254837475357, −8.276911895899791172137384170713, −6.99244850594775641708476934736, −4.37742455486164328761981449108, −3.77792064457441427014083380539, −2.04859868907675581650995899585,
0.21030744653775516846655122657, 1.71783164558578473133297397705, 3.09718009268942359849182785808, 5.06743510926847890243346793662, 6.47156239331448620788643065646, 8.311901461894302737167807047891, 9.212154440367386356237567623329, 10.62002563956613514908130527719, 12.58127377865421462491688527886, 13.43189344608350551498973227791