| L(s) = 1 | + (5.45 + 30.9i)2-s + (996. − 362. i)4-s + (−2.76e3 − 2.32e3i)5-s + (4.72e4 + 1.71e4i)7-s + (4.88e4 + 8.45e4i)8-s + (5.68e4 − 9.83e4i)10-s + (−6.42e4 + 5.39e4i)11-s + (−2.83e5 + 1.60e6i)13-s + (−2.74e5 + 1.55e6i)14-s + (−6.87e5 + 5.76e5i)16-s + (3.66e6 − 6.35e6i)17-s + (−2.44e5 − 4.24e5i)19-s + (−3.60e6 − 1.31e6i)20-s + (−2.01e6 − 1.69e6i)22-s + (−2.68e7 + 9.76e6i)23-s + ⋯ |
| L(s) = 1 | + (0.120 + 0.683i)2-s + (0.486 − 0.177i)4-s + (−0.396 − 0.332i)5-s + (1.06 + 0.386i)7-s + (0.526 + 0.912i)8-s + (0.179 − 0.311i)10-s + (−0.120 + 0.100i)11-s + (−0.211 + 1.20i)13-s + (−0.136 + 0.772i)14-s + (−0.163 + 0.137i)16-s + (0.626 − 1.08i)17-s + (−0.0226 − 0.0393i)19-s + (−0.251 − 0.0916i)20-s + (−0.0835 − 0.0700i)22-s + (−0.869 + 0.316i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.98585 + 2.09498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98585 + 2.09498i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-5.45 - 30.9i)T + (-1.92e3 + 700. i)T^{2} \) |
| 5 | \( 1 + (2.76e3 + 2.32e3i)T + (8.47e6 + 4.80e7i)T^{2} \) |
| 7 | \( 1 + (-4.72e4 - 1.71e4i)T + (1.51e9 + 1.27e9i)T^{2} \) |
| 11 | \( 1 + (6.42e4 - 5.39e4i)T + (4.95e10 - 2.80e11i)T^{2} \) |
| 13 | \( 1 + (2.83e5 - 1.60e6i)T + (-1.68e12 - 6.12e11i)T^{2} \) |
| 17 | \( 1 + (-3.66e6 + 6.35e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.44e5 + 4.24e5i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.68e7 - 9.76e6i)T + (7.29e14 - 6.12e14i)T^{2} \) |
| 29 | \( 1 + (-3.21e7 - 1.82e8i)T + (-1.14e16 + 4.17e15i)T^{2} \) |
| 31 | \( 1 + (-2.54e8 + 9.27e7i)T + (1.94e16 - 1.63e16i)T^{2} \) |
| 37 | \( 1 + (3.23e8 - 5.60e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (3.38e6 - 1.92e7i)T + (-5.17e17 - 1.88e17i)T^{2} \) |
| 43 | \( 1 + (-7.34e8 + 6.16e8i)T + (1.61e17 - 9.15e17i)T^{2} \) |
| 47 | \( 1 + (-5.71e8 - 2.07e8i)T + (1.89e18 + 1.58e18i)T^{2} \) |
| 53 | \( 1 - 1.99e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-2.28e9 - 1.91e9i)T + (5.23e18 + 2.96e19i)T^{2} \) |
| 61 | \( 1 + (3.02e9 + 1.10e9i)T + (3.33e19 + 2.79e19i)T^{2} \) |
| 67 | \( 1 + (2.40e9 - 1.36e10i)T + (-1.14e20 - 4.17e19i)T^{2} \) |
| 71 | \( 1 + (1.04e10 - 1.81e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 + (-1.12e9 - 1.95e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-6.93e8 - 3.93e9i)T + (-7.02e20 + 2.55e20i)T^{2} \) |
| 83 | \( 1 + (-2.59e9 - 1.47e10i)T + (-1.21e21 + 4.40e20i)T^{2} \) |
| 89 | \( 1 + (-7.02e9 - 1.21e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (7.33e10 - 6.15e10i)T + (1.24e21 - 7.04e21i)T^{2} \) |
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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
−11.92745152526956386349941392292, −11.67529044302018510850146246594, −10.23131547223094373233363666116, −8.700613281490595432861751764108, −7.76788585975428389208160483614, −6.74092113829605276139282743824, −5.36184918064696969724030668909, −4.48264494365551581646280615678, −2.45747460659257812896641529207, −1.25848492189840203323844415974,
0.72173702417141262691257515879, 1.96180741800528108988399543217, 3.23720796242641870424502709335, 4.34827091788025873543978874583, 5.97703214931777473531805283792, 7.54503965935170248562650047014, 8.122467603477688799527505002796, 10.16502548511651580304293174805, 10.76121718827444530627366756680, 11.76454343098732828178085293637
