| L(s) = 1 | + (112. + 81.5i)2-s + (22.6 − 69.8i)3-s + (3.41e3 + 1.05e4i)4-s + (3.90e4 − 2.83e4i)5-s + (8.24e3 − 5.99e3i)6-s + (1.28e5 + 3.95e5i)7-s + (−1.23e5 + 3.79e5i)8-s + (1.28e6 + 9.33e5i)9-s + 6.69e6·10-s + (−4.92e6 + 3.20e6i)11-s + 8.12e5·12-s + (−1.58e7 − 1.15e7i)13-s + (−1.78e7 + 5.48e7i)14-s + (−1.09e6 − 3.37e6i)15-s + (2.85e7 − 2.07e7i)16-s + (4.93e7 − 3.58e7i)17-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.901i)2-s + (0.0179 − 0.0553i)3-s + (0.417 + 1.28i)4-s + (1.11 − 0.811i)5-s + (0.0721 − 0.0524i)6-s + (0.412 + 1.26i)7-s + (−0.166 + 0.511i)8-s + (0.806 + 0.585i)9-s + 2.11·10-s + (−0.838 + 0.545i)11-s + 0.0785·12-s + (−0.910 − 0.661i)13-s + (−0.632 + 1.94i)14-s + (−0.0248 − 0.0764i)15-s + (0.425 − 0.309i)16-s + (0.495 − 0.360i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.33046 + 2.34889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.33046 + 2.34889i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.92e6 - 3.20e6i)T \) |
| good | 2 | \( 1 + (-112. - 81.5i)T + (2.53e3 + 7.79e3i)T^{2} \) |
| 3 | \( 1 + (-22.6 + 69.8i)T + (-1.28e6 - 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-3.90e4 + 2.83e4i)T + (3.77e8 - 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-1.28e5 - 3.95e5i)T + (-7.83e10 + 5.69e10i)T^{2} \) |
| 13 | \( 1 + (1.58e7 + 1.15e7i)T + (9.35e13 + 2.88e14i)T^{2} \) |
| 17 | \( 1 + (-4.93e7 + 3.58e7i)T + (3.06e15 - 9.41e15i)T^{2} \) |
| 19 | \( 1 + (3.82e7 - 1.17e8i)T + (-3.40e16 - 2.47e16i)T^{2} \) |
| 23 | \( 1 - 5.54e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.26e9 + 3.89e9i)T + (-8.30e18 + 6.03e18i)T^{2} \) |
| 31 | \( 1 + (5.28e9 + 3.83e9i)T + (7.54e18 + 2.32e19i)T^{2} \) |
| 37 | \( 1 + (5.33e9 + 1.64e10i)T + (-1.97e20 + 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-3.94e9 + 1.21e10i)T + (-7.48e20 - 5.43e20i)T^{2} \) |
| 43 | \( 1 + 4.69e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.40e10 - 7.38e10i)T + (-4.41e21 - 3.20e21i)T^{2} \) |
| 53 | \( 1 + (1.32e11 + 9.65e10i)T + (8.04e21 + 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-2.53e10 - 7.80e10i)T + (-8.49e22 + 6.17e22i)T^{2} \) |
| 61 | \( 1 + (2.35e11 - 1.71e11i)T + (5.00e22 - 1.53e23i)T^{2} \) |
| 67 | \( 1 - 7.01e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-1.22e12 + 8.86e11i)T + (3.60e23 - 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-5.67e11 - 1.74e12i)T + (-1.35e24 + 9.82e23i)T^{2} \) |
| 79 | \( 1 + (3.34e12 + 2.43e12i)T + (1.44e24 + 4.43e24i)T^{2} \) |
| 83 | \( 1 + (2.24e12 - 1.62e12i)T + (2.74e24 - 8.43e24i)T^{2} \) |
| 89 | \( 1 - 6.23e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (2.42e11 + 1.76e11i)T + (2.07e25 + 6.40e25i)T^{2} \) |
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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
−17.11781251453602034096397748396, −15.76999837374010181772470391727, −14.69026241243363071435764539331, −13.15104496169018488811113158243, −12.51088937203209509108980728559, −9.748423244727791228995697826725, −7.69481925780888465518430000041, −5.60471845490578379485383358008, −4.96454616580197841361965649039, −2.16669752165207820121195757436,
1.61415291870014999101468445485, 3.27685834329163244870529795352, 4.96419877745668968909428650235, 6.89717934714895902086524559938, 10.07571575081514088861475533914, 10.96704416159926859081494510358, 12.84399586481983283947130356751, 13.87570179418944196949361262885, 14.77020705171796441659677936485, 17.06886388521629375437971294802
