| L(s) = 1 | − 58.7i·2-s − 301. i·3-s − 2.43e3·4-s − 2.78e3i·5-s − 1.77e4·6-s + 4.09e3i·7-s + 8.27e4i·8-s − 3.18e4·9-s − 1.63e5·10-s − 1.23e5·11-s + 7.33e5i·12-s + 3.47e5·13-s + 2.40e5·14-s − 8.38e5·15-s + 2.37e6·16-s − 8.25e4·17-s + ⋯ |
| L(s) = 1 | − 1.83i·2-s − 1.24i·3-s − 2.37·4-s − 0.889i·5-s − 2.27·6-s + 0.243i·7-s + 2.52i·8-s − 0.539·9-s − 1.63·10-s − 0.767·11-s + 2.94i·12-s + 0.935·13-s + 0.447·14-s − 1.10·15-s + 2.26·16-s − 0.0581·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0406i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.999 + 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.542145 - 0.0110360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.542145 - 0.0110360i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.46e8 + 5.98e6i)T \) |
| good | 2 | \( 1 + 58.7iT - 1.02e3T^{2} \) |
| 3 | \( 1 + 301. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 2.78e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 4.09e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.23e5T + 2.59e10T^{2} \) |
| 13 | \( 1 - 3.47e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 8.25e4T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.73e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 1.02e7T + 4.14e13T^{2} \) |
| 29 | \( 1 + 6.06e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 4.26e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 1.00e8iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 9.51e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 6.18e6T + 5.25e16T^{2} \) |
| 53 | \( 1 + 3.72e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 7.30e8T + 5.11e17T^{2} \) |
| 61 | \( 1 - 5.55e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 + 1.30e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 2.26e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.19e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 5.10e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 7.51e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 5.82e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 8.05e9T + 7.37e19T^{2} \) |
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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
−12.60800767824383367724359691402, −11.67038209973287533286545825641, −10.38625053087392653865589556616, −8.973100946214492308352026820026, −7.937340581778786160595337321598, −5.64370666633314046110523613076, −3.96433305317155919118450808455, −2.23691584454939448665750359606, −1.32816977273461670570634966865, −0.18892030775990726682886710679,
3.57736054548185332810493758952, 4.80482852234585374804274585974, 6.06682666019680176286090132326, 7.31064996136421672151056520231, 8.599817437265141457315569179398, 9.855819969031708819787994987798, 10.87167811096986515589922932621, 13.26440122067668959905152536609, 14.31570554729738175568309429352, 15.19448643866463877365790100558
