| L(s) = 1 | + (15.6 − 27.0i)2-s + (−56.7 − 98.3i)3-s + (−231. − 401. i)4-s + (−239. + 415. i)5-s − 3.54e3·6-s + (1.42e3 − 6.18e3i)7-s + 1.50e3·8-s + (3.39e3 − 5.88e3i)9-s + (7.48e3 + 1.29e4i)10-s + (3.36e4 + 5.82e4i)11-s + (−2.63e4 + 4.55e4i)12-s + 3.96e4·13-s + (−1.45e5 − 1.35e5i)14-s + 5.44e4·15-s + (1.42e5 − 2.46e5i)16-s + (−1.06e5 − 1.84e5i)17-s + ⋯ |
| L(s) = 1 | + (0.690 − 1.19i)2-s + (−0.404 − 0.700i)3-s + (−0.452 − 0.784i)4-s + (−0.171 + 0.297i)5-s − 1.11·6-s + (0.224 − 0.974i)7-s + 0.130·8-s + (0.172 − 0.298i)9-s + (0.236 + 0.410i)10-s + (0.692 + 1.19i)11-s + (−0.366 + 0.634i)12-s + 0.384·13-s + (−1.00 − 0.941i)14-s + 0.277·15-s + (0.542 − 0.939i)16-s + (−0.309 − 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.780246 - 1.64926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.780246 - 1.64926i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.42e3 + 6.18e3i)T \) |
| good | 2 | \( 1 + (-15.6 + 27.0i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (56.7 + 98.3i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (239. - 415. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-3.36e4 - 5.82e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 3.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.06e5 + 1.84e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (5.66e5 - 9.81e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-3.65e5 + 6.32e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 3.83e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.73e6 - 4.73e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-3.83e6 + 6.64e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.08e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.92e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.11e7 + 3.65e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.68e7 - 4.64e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (9.06e6 + 1.57e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.48e7 - 2.56e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.34e7 - 1.44e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 3.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.51e7 - 2.61e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.62e8 - 4.53e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 6.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-4.80e8 + 8.32e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 3.75e8T + 7.60e17T^{2} \) |
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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
−20.04613738447955242617280209916, −18.61189706326003169660074914784, −17.05953819236712675019433860517, −14.50011559783591668337512871295, −12.94008171622468572567545562869, −11.84589035694933335132477107806, −10.34985185294281185073826999222, −7.05709208054214529132260120731, −4.06763266954747862157287758756, −1.44004437017363897511066140888,
4.60343439957740483311392502826, 6.13954108144853489870871027378, 8.583999431897183960752525013986, 11.16160685548325804221917212680, 13.34345440394659071981071455218, 15.07426065830085932308228818436, 15.99397392162567632541720090948, 17.13410727415554242540554299988, 19.29151334065450013285118221877, 21.60384498213504779769264721961
