| L(s) = 1 | + (10.0 + 5.17i)2-s + (−24.5 + 59.2i)3-s + (74.5 + 104. i)4-s + (−27.7 + 11.4i)5-s + (−553. + 469. i)6-s + (−102. + 102. i)7-s + (211. + 1.43e3i)8-s + (−1.36e3 − 1.36e3i)9-s + (−338. − 27.7i)10-s + (−743. − 1.79e3i)11-s + (−7.99e3 + 1.86e3i)12-s + (−7.82e3 − 3.24e3i)13-s + (−1.55e3 + 500. i)14-s − 1.92e3i·15-s + (−5.27e3 + 1.55e4i)16-s + 7.09e3i·17-s + ⋯ |
| L(s) = 1 | + (0.889 + 0.457i)2-s + (−0.524 + 1.26i)3-s + (0.582 + 0.812i)4-s + (−0.0991 + 0.0410i)5-s + (−1.04 + 0.887i)6-s + (−0.112 + 0.112i)7-s + (0.146 + 0.989i)8-s + (−0.623 − 0.623i)9-s + (−0.106 − 0.00878i)10-s + (−0.168 − 0.406i)11-s + (−1.33 + 0.311i)12-s + (−0.988 − 0.409i)13-s + (−0.151 + 0.0487i)14-s − 0.147i·15-s + (−0.321 + 0.946i)16-s + 0.350i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.215802 + 2.01584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215802 + 2.01584i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.0 - 5.17i)T \) |
| good | 3 | \( 1 + (24.5 - 59.2i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (27.7 - 11.4i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (102. - 102. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (743. + 1.79e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (7.82e3 + 3.24e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 7.09e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.99e4 - 1.24e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.91e4 - 4.91e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.92e3 - 4.64e3i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 2.93e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-2.08e5 + 8.62e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-3.16e5 - 3.16e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (1.81e5 + 4.37e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.82e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (3.27e5 + 7.90e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (9.43e5 - 3.90e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (7.77e5 - 1.87e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-6.28e5 + 1.51e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-3.98e6 + 3.98e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (3.84e6 + 3.84e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 2.95e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-6.76e6 - 2.80e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (1.99e6 - 1.99e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.33e7T + 8.07e13T^{2} \) |
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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
−15.65369753017874663499269312250, −14.96767274508080023528808544232, −13.55574498464444050299087797040, −12.07346937458648539338738694109, −10.98455069320466582008840537581, −9.612298072643572469433958218691, −7.68589797153308891278063747528, −5.80440809245706204400242964831, −4.74015421731372275347588021371, −3.25003628697949700373000398392,
0.77215569983722697324505013961, 2.46792082933786724733475294459, 4.79821100173119705431807381326, 6.41210934911119763565786727529, 7.44011323896284518057205420518, 9.870713934388639304290969315398, 11.51939732039814370271157898753, 12.26120267443444144376348427375, 13.24032886506554063019690719766, 14.29259377214353325016383265835
