| L(s) = 1 | + (12 − 20.7i)2-s + (126 + 218. i)3-s + (736 + 1.27e3i)4-s + (2.41e3 − 4.18e3i)5-s + 6.04e3·6-s + 8.44e4·8-s + (5.68e4 − 9.84e4i)9-s + (−5.79e4 − 1.00e5i)10-s + (−2.67e5 − 4.62e5i)11-s + (−1.85e5 + 3.21e5i)12-s + 5.77e5·13-s + 1.21e6·15-s + (−4.93e5 + 8.54e5i)16-s + (−3.45e6 − 5.98e6i)17-s + (−1.36e6 − 2.36e6i)18-s + (5.33e6 − 9.23e6i)19-s + ⋯ |
| L(s) = 1 | + (0.265 − 0.459i)2-s + (0.299 + 0.518i)3-s + (0.359 + 0.622i)4-s + (0.345 − 0.598i)5-s + 0.317·6-s + 0.911·8-s + (0.320 − 0.555i)9-s + (−0.183 − 0.317i)10-s + (−0.500 − 0.866i)11-s + (−0.215 + 0.372i)12-s + 0.431·13-s + 0.413·15-s + (−0.117 + 0.203i)16-s + (−0.589 − 1.02i)17-s + (−0.170 − 0.294i)18-s + (0.493 − 0.855i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.03384 - 1.27143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.03384 - 1.27143i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-12 + 20.7i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-126 - 218. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-2.41e3 + 4.18e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (2.67e5 + 4.62e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 5.77e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (3.45e6 + 5.98e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-5.33e6 + 9.23e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (9.32e6 - 1.61e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.28e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (2.64e7 + 4.57e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-9.11e7 + 1.57e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 3.08e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.71e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.34e9 + 2.32e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-7.98e8 - 1.38e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.59e9 + 4.49e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.47e9 + 6.02e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-7.74e9 - 1.34e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 9.79e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-7.31e8 - 1.26e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.90e10 - 3.30e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.93e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.24e10 - 2.16e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 7.50e10T + 7.15e21T^{2} \) |
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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
−13.11652284482993564658038988533, −11.89659023202240636566138197641, −10.86739919228157851379995069197, −9.464609138985420190652111133750, −8.425552420259511262657977813724, −6.90012787818593900548102645527, −5.10083608107666506593453654550, −3.76003447954918115558626209742, −2.63266405292909951432822305995, −0.900005096472230294746675404874,
1.43981670022813946246416561018, 2.44655220540359277797802574558, 4.58666803203367537256189073990, 6.07292621766043728724281557183, 7.02550027721407422630161078079, 8.149083239662200641081063194734, 10.11319766071529462204154800299, 10.71780581200859915008162222552, 12.46723612742482659933225337368, 13.63127373498507838950278919510
