| L(s) = 1 | + (121.5 − 210. i)3-s + (−3.94e3 − 6.83e3i)5-s + (3.91e4 − 2.10e4i)7-s + (−2.95e4 − 5.11e4i)9-s + (−1.63e5 + 2.83e5i)11-s − 1.91e6·13-s − 1.91e6·15-s + (7.77e5 − 1.34e6i)17-s + (−4.53e6 − 7.86e6i)19-s + (3.20e5 − 1.08e7i)21-s + (−3.15e6 − 5.46e6i)23-s + (−6.71e6 + 1.16e7i)25-s − 1.43e7·27-s − 1.35e7·29-s + (−4.74e7 + 8.21e7i)31-s + ⋯ |
| L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.564 − 0.977i)5-s + (0.880 − 0.474i)7-s + (−0.166 − 0.288i)9-s + (−0.306 + 0.530i)11-s − 1.43·13-s − 0.651·15-s + (0.132 − 0.230i)17-s + (−0.420 − 0.728i)19-s + (0.0171 − 0.577i)21-s + (−0.102 − 0.177i)23-s + (−0.137 + 0.238i)25-s − 0.192·27-s − 0.122·29-s + (−0.297 + 0.515i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0971349 + 0.151510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0971349 + 0.151510i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-121.5 + 210. i)T \) |
| 7 | \( 1 + (-3.91e4 + 2.10e4i)T \) |
| good | 5 | \( 1 + (3.94e3 + 6.83e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.63e5 - 2.83e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.91e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-7.77e5 + 1.34e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (4.53e6 + 7.86e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (3.15e6 + 5.46e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.35e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (4.74e7 - 8.21e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.77e8 - 4.80e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 2.48e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.69e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (8.91e8 + 1.54e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (4.82e8 - 8.35e8i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.65e9 - 4.59e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.16e9 - 5.48e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.76e9 - 4.79e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 9.74e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (3.97e9 - 6.87e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-6.70e9 - 1.16e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.49e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.93e10 - 3.34e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.12e11T + 7.15e21T^{2} \) |
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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
−11.57156931778049894156614359657, −10.13392341497136991480402325672, −8.822061582005103676175284128075, −7.86788926645127666998012842392, −7.05273195290975398084576621767, −5.09473960546727728005682918497, −4.35022996206398874878750557959, −2.49933387124864340018805036810, −1.19812583088249773038130152614, −0.04304106017020957591427680283,
2.07891206508043129527505859256, 3.18019166598417770399597658405, 4.48594388251555489879827440236, 5.76642934899189395010933444487, 7.39706972178242041398571739211, 8.175840522470014556089369744940, 9.548300373428402264835874921838, 10.71501811327130772960749828683, 11.45262893725795290185004559154, 12.60987260878237656447118969442
