| L(s) = 1 | + (−45.3 + 139. i)2-s + (1.09e3 − 794. i)3-s + (−1.07e4 − 7.83e3i)4-s + (−6.34e3 − 1.95e4i)5-s + (6.12e4 + 1.88e5i)6-s + (3.63e5 + 2.64e5i)7-s + (6.08e5 − 4.42e5i)8-s + (7.12e4 − 2.19e5i)9-s + 3.00e6·10-s + (5.69e6 + 1.44e6i)11-s − 1.79e7·12-s + (−6.39e6 + 1.96e7i)13-s + (−5.33e7 + 3.87e7i)14-s + (−2.24e7 − 1.62e7i)15-s + (3.85e5 + 1.18e6i)16-s + (2.27e7 + 6.99e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.500 + 1.54i)2-s + (0.865 − 0.628i)3-s + (−1.31 − 0.955i)4-s + (−0.181 − 0.558i)5-s + (0.535 + 1.64i)6-s + (1.16 + 0.849i)7-s + (0.821 − 0.596i)8-s + (0.0446 − 0.137i)9-s + 0.951·10-s + (0.969 + 0.246i)11-s − 1.73·12-s + (−0.367 + 1.13i)13-s + (−1.89 + 1.37i)14-s + (−0.508 − 0.369i)15-s + (0.00574 + 0.0176i)16-s + (0.228 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.987374 + 1.53105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987374 + 1.53105i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-5.69e6 - 1.44e6i)T \) |
| good | 2 | \( 1 + (45.3 - 139. i)T + (-6.62e3 - 4.81e3i)T^{2} \) |
| 3 | \( 1 + (-1.09e3 + 794. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (6.34e3 + 1.95e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.63e5 - 2.64e5i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (6.39e6 - 1.96e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-2.27e7 - 6.99e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-1.00e8 + 7.27e7i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 + 2.65e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-2.12e9 - 1.54e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (2.27e9 - 6.99e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (3.54e9 + 2.57e9i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-2.24e10 + 1.63e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 + 6.71e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-6.28e10 + 4.56e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-8.28e10 + 2.55e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (4.78e11 + 3.47e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-7.46e10 - 2.29e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 - 8.77e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-1.53e11 - 4.72e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (1.32e12 + 9.62e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-8.22e10 + 2.53e11i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (1.19e12 + 3.66e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 - 4.59e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-1.05e12 + 3.24e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−17.54865827449713357134939566590, −16.29431275326494750354695287317, −14.73549558367000520005870864519, −14.14298567514985815650923831764, −12.05374933971631046094398258108, −8.964810342062515897811876990605, −8.342259303837273715308466288920, −6.94465493935983588073110528354, −4.99129921951862344251586441357, −1.66958820273030729605739134473,
0.992365804756581764843564966955, 2.88610254660860745454245523576, 4.09174313103100617150314955675, 7.957319925022279000117680909966, 9.489182144459912273639379696699, 10.66257930720938895839688653762, 11.82797258612313929894300922392, 13.87707281766824939597817008386, 14.97980116734516254702243327100, 17.30027806058117550616708335445
