| L(s) = 1 | + (16 + 27.7i)2-s + (−158. + 274. i)3-s + (−511. + 886. i)4-s + (−6.00e3 − 1.04e4i)5-s − 1.01e4·6-s + (3.41e4 − 2.84e4i)7-s − 3.27e4·8-s + (3.84e4 + 6.65e4i)9-s + (1.92e5 − 3.33e5i)10-s + (1.87e5 − 3.24e5i)11-s + (−1.62e5 − 2.80e5i)12-s − 6.25e5·13-s + (1.33e6 + 4.91e5i)14-s + 3.80e6·15-s + (−5.24e5 − 9.08e5i)16-s + (3.56e6 − 6.16e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.376 + 0.651i)3-s + (−0.249 + 0.433i)4-s + (−0.860 − 1.48i)5-s − 0.531·6-s + (0.768 − 0.639i)7-s − 0.353·8-s + (0.216 + 0.375i)9-s + (0.608 − 1.05i)10-s + (0.350 − 0.606i)11-s + (−0.188 − 0.325i)12-s − 0.467·13-s + (0.663 + 0.244i)14-s + 1.29·15-s + (−0.125 − 0.216i)16-s + (0.608 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.07880 - 0.547796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07880 - 0.547796i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 - 27.7i)T \) |
| 7 | \( 1 + (-3.41e4 + 2.84e4i)T \) |
| good | 3 | \( 1 + (158. - 274. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (6.00e3 + 1.04e4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.87e5 + 3.24e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 6.25e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-3.56e6 + 6.16e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (9.63e6 + 1.66e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.19e7 + 2.07e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 7.06e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (7.36e7 - 1.27e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.68e8 + 4.64e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 4.82e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.66e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.35e8 - 9.27e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.88e9 - 4.99e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.69e9 + 2.92e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.55e9 + 4.42e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.24e9 - 2.14e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.40e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-9.20e8 + 1.59e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.37e10 - 2.38e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 3.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.01e10 + 1.75e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.43e11T + 7.15e21T^{2} \) |
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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
−16.46246547773560603286977880906, −15.70530733152513838340121314783, −14.01404644471561009365332983435, −12.48760065135118101980531482009, −11.08654490765395400358079833349, −8.943137753182409307195334054509, −7.58272851727136452326165298810, −5.02836229943908090349208410838, −4.32109984959204031239102045699, −0.53338866246804771731124828469,
1.87480216840658501293542927807, 3.80709711053136351382600233268, 6.21585976389065476653017788690, 7.76941613459852179842938038895, 10.27107328403780872175710944428, 11.66912741439102726982137887524, 12.38039272844645281604106380331, 14.55062963462616608757380186649, 15.10852457700435703590799341156, 17.58871855781836477599985202866
