| L(s) = 1 | + (−87.3 + 50.4i)2-s + (197. + 342. i)3-s + (994. − 1.72e3i)4-s − 4.30e3i·5-s + (−3.45e4 − 1.99e4i)6-s + (4.11e5 + 2.37e5i)7-s − 6.25e5i·8-s + (7.19e5 − 1.24e6i)9-s + (2.16e5 + 3.75e5i)10-s + (1.06e6 − 6.16e5i)11-s + 7.85e5·12-s + (1.29e7 + 1.15e7i)13-s − 4.78e7·14-s + (1.47e6 − 8.49e5i)15-s + (3.97e7 + 6.88e7i)16-s + (−9.49e7 + 1.64e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.965 + 0.557i)2-s + (0.156 + 0.270i)3-s + (0.121 − 0.210i)4-s − 0.123i·5-s + (−0.302 − 0.174i)6-s + (1.32 + 0.762i)7-s − 0.844i·8-s + (0.451 − 0.781i)9-s + (0.0686 + 0.118i)10-s + (0.181 − 0.104i)11-s + 0.0759·12-s + (0.745 + 0.666i)13-s − 1.69·14-s + (0.0333 − 0.0192i)15-s + (0.591 + 1.02i)16-s + (−0.954 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.723792 + 0.986816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.723792 + 0.986816i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.29e7 - 1.15e7i)T \) |
| good | 2 | \( 1 + (87.3 - 50.4i)T + (4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-197. - 342. i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + 4.30e3iT - 1.22e9T^{2} \) |
| 7 | \( 1 + (-4.11e5 - 2.37e5i)T + (4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-1.06e6 + 6.16e5i)T + (1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (9.49e7 - 1.64e8i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.67e8 + 9.68e7i)T + (2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-1.98e8 - 3.43e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + (1.75e9 + 3.04e9i)T + (-5.13e18 + 8.88e18i)T^{2} \) |
| 31 | \( 1 - 7.47e9iT - 2.44e19T^{2} \) |
| 37 | \( 1 + (3.58e8 - 2.06e8i)T + (1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (-3.32e10 + 1.92e10i)T + (4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (2.46e10 - 4.27e10i)T + (-8.59e20 - 1.48e21i)T^{2} \) |
| 47 | \( 1 - 8.19e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 + 1.31e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + (-5.73e10 - 3.31e10i)T + (5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-1.77e11 + 3.07e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-1.44e10 + 8.31e9i)T + (2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (5.99e11 + 3.46e11i)T + (5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 - 2.10e12iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 1.84e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.15e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + (5.87e11 - 3.39e11i)T + (1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (-7.75e12 - 4.47e12i)T + (3.36e25 + 5.82e25i)T^{2} \) |
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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
−17.28654755769046185608921852192, −15.71783454091663863663457454979, −14.78641649289062501549582518319, −12.72450091122127947438038640095, −10.98138365055491613143116192434, −9.081912849377943649751460082178, −8.375595655541302104186057150473, −6.50003361218116457034182675456, −4.19277469176002754106260382250, −1.43096792772208055083576211556,
0.792271491201526890586042459192, 2.10235486743919320423784424153, 4.80459008932169756735120280039, 7.46908965873218797203088554805, 8.682642320243457066925228952979, 10.49689948482709798278399858618, 11.27021961084955240416152220629, 13.42757337593805182566592531303, 14.66861975084683070903616593757, 16.66500086034633151982143243311
