L(s) = 1 | + (−2.65e5 − 1.53e5i)3-s + (−4.19e6 + 7.26e6i)4-s + (4.93e9 + 1.74e9i)7-s + (4.70e10 + 8.15e10i)9-s + (2.22e12 − 1.28e12i)12-s + 8.36e12i·13-s + (−3.51e13 − 6.09e13i)16-s + (7.49e14 − 4.32e14i)19-s + (−1.04e15 − 1.21e15i)21-s + (5.96e15 − 1.03e16i)25-s + (2 − 2.88e16i)27-s + (−3.33e16 + 2.85e16i)28-s + (−2.44e17 − 1.41e17i)31-s − 7.89e17·36-s + (1.02e18 + 1.77e18i)37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.500i)3-s + (−0.5 + 0.866i)4-s + (0.942 + 0.333i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + 1.29i·13-s + (−0.499 − 0.866i)16-s + (1.47 − 0.852i)19-s + (−0.650 − 0.759i)21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.759 + 0.650i)28-s + (−1.72 − 0.997i)31-s − 0.999·36-s + (0.948 + 1.64i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(\approx\) |
\(1.407141595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407141595\) |
\(L(\frac{25}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.65e5 + 1.53e5i)T \) |
| 7 | \( 1 + (-4.93e9 - 1.74e9i)T \) |
good | 2 | \( 1 + (4.19e6 - 7.26e6i)T^{2} \) |
| 5 | \( 1 + (-5.96e15 + 1.03e16i)T^{2} \) |
| 11 | \( 1 + (4.47e23 + 7.75e23i)T^{2} \) |
| 13 | \( 1 - 8.36e12iT - 4.17e25T^{2} \) |
| 17 | \( 1 + (-9.98e27 - 1.72e28i)T^{2} \) |
| 19 | \( 1 + (-7.49e14 + 4.32e14i)T + (1.28e29 - 2.23e29i)T^{2} \) |
| 23 | \( 1 + (1.04e31 - 1.80e31i)T^{2} \) |
| 29 | \( 1 - 4.31e33T^{2} \) |
| 31 | \( 1 + (2.44e17 + 1.41e17i)T + (1.00e34 + 1.73e34i)T^{2} \) |
| 37 | \( 1 + (-1.02e18 - 1.77e18i)T + (-5.85e35 + 1.01e36i)T^{2} \) |
| 41 | \( 1 + 1.24e37T^{2} \) |
| 43 | \( 1 - 5.06e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + (-1.43e38 + 2.48e38i)T^{2} \) |
| 53 | \( 1 + (2.27e39 + 3.94e39i)T^{2} \) |
| 59 | \( 1 + (-2.68e40 - 4.64e40i)T^{2} \) |
| 61 | \( 1 + (-5.35e20 + 3.09e20i)T + (5.77e40 - 1.00e41i)T^{2} \) |
| 67 | \( 1 + (9.45e20 - 1.63e21i)T + (-4.99e41 - 8.65e41i)T^{2} \) |
| 71 | \( 1 - 3.79e42T^{2} \) |
| 73 | \( 1 + (-4.45e21 - 2.57e21i)T + (3.59e42 + 6.22e42i)T^{2} \) |
| 79 | \( 1 + (1.05e20 + 1.82e20i)T + (-2.21e43 + 3.82e43i)T^{2} \) |
| 83 | \( 1 + 1.37e44T^{2} \) |
| 89 | \( 1 + (-3.42e44 + 5.93e44i)T^{2} \) |
| 97 | \( 1 - 1.10e23iT - 4.96e45T^{2} \) |
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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
−13.16694564848581433659405089647, −11.89691570558643866897033074777, −11.30177275228259221800933341925, −9.340750209581474426794812390089, −7.987984353548151344800905300434, −6.92519154502220729227694761671, −5.25960390992068595331298672927, −4.26561649546872621750489968457, −2.37168952741411211504883962441, −0.952983901044094813043617832249,
0.52437859922303641671903908166, 1.40314194551751284386810968994, 3.71990128892207607612322580791, 5.12364536369743120763734176788, 5.66014813535109076661070777147, 7.49224354736094772146058847012, 9.222832300599903201784964751967, 10.41672861414817540722457325835, 11.16148758290273052284797810705, 12.69384567445292265171837487965