| L(s) = 1 | + (−21.0 + 36.3i)2-s + (205. + 355. i)3-s + (141. + 245. i)4-s + (−41.2 + 71.4i)5-s − 1.72e4·6-s + (−4.33e4 + 1.00e4i)7-s − 9.79e4·8-s + (4.34e3 − 7.53e3i)9-s + (−1.73e3 − 3.00e3i)10-s + (1.60e5 + 2.78e5i)11-s + (−5.81e4 + 1.00e5i)12-s + 8.11e5·13-s + (5.44e5 − 1.78e6i)14-s − 3.38e4·15-s + (1.76e6 − 3.05e6i)16-s + (2.13e6 + 3.69e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.464 + 0.803i)2-s + (0.487 + 0.844i)3-s + (0.0692 + 0.119i)4-s + (−0.00590 + 0.0102i)5-s − 0.905·6-s + (−0.974 + 0.225i)7-s − 1.05·8-s + (0.0245 − 0.0425i)9-s + (−0.00548 − 0.00949i)10-s + (0.300 + 0.521i)11-s + (−0.0675 + 0.116i)12-s + 0.606·13-s + (0.270 − 0.887i)14-s − 0.0115·15-s + (0.421 − 0.729i)16-s + (0.364 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.142102 + 1.21645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.142102 + 1.21645i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.33e4 - 1.00e4i)T \) |
| good | 2 | \( 1 + (21.0 - 36.3i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-205. - 355. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (41.2 - 71.4i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.60e5 - 2.78e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 8.11e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.13e6 - 3.69e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (7.49e6 - 1.29e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.29e6 + 5.70e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.83e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (7.42e7 + 1.28e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.93e8 - 5.07e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 7.84e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-7.11e8 + 1.23e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.68e8 + 2.92e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (5.07e9 + 8.79e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.96e9 + 3.39e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-8.62e9 - 1.49e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (7.04e9 + 1.21e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (3.25e9 - 5.63e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 1.57e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.14e10 + 7.18e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 6.77e10T + 7.15e21T^{2} \) |
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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
−20.49883238545711762459193329628, −18.72873307932535808179933733448, −16.97196311697675525620343331084, −15.83166779251143283489379785608, −14.83246682010766401994312581709, −12.47816204088464352344983144395, −9.924572105384588291393034344621, −8.562515899265089739146815352991, −6.46914314793930104137336612401, −3.50160656694902905199538595954,
0.847289427902800774780190116561, 2.77784466053276236280323956056, 6.64028393766974782408402709029, 8.894792706372553045282475351505, 10.64832756738846788769837142874, 12.46257824905821630059921542734, 13.90118362140175887683279150517, 15.98788050306776545044346110827, 18.12423200460619562097144573849, 19.29737863219414639289287090753
