| L(s) = 1 | + 5.51i·2-s + 292. i·3-s + 993.·4-s − 3.20e3i·5-s − 1.61e3·6-s + 1.65e4i·7-s + 1.11e4i·8-s − 2.67e4·9-s + 1.76e4·10-s + 3.15e4·11-s + 2.90e5i·12-s + 2.40e5·13-s − 9.10e4·14-s + 9.38e5·15-s + 9.56e5·16-s + 1.02e6·17-s + ⋯ |
| L(s) = 1 | + 0.172i·2-s + 1.20i·3-s + 0.970·4-s − 1.02i·5-s − 0.207·6-s + 0.982i·7-s + 0.339i·8-s − 0.452·9-s + 0.176·10-s + 0.196·11-s + 1.16i·12-s + 0.646·13-s − 0.169·14-s + 1.23·15-s + 0.911·16-s + 0.723·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.64834 + 2.01307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64834 + 2.01307i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.90e7 - 1.44e8i)T \) |
| good | 2 | \( 1 - 5.51iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 292. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 3.20e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 1.65e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 3.15e4T + 2.59e10T^{2} \) |
| 13 | \( 1 - 2.40e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 1.02e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.67e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 6.08e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 1.23e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 5.08e6T + 8.19e14T^{2} \) |
| 37 | \( 1 - 2.30e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 7.74e7T + 1.34e16T^{2} \) |
| 47 | \( 1 + 2.22e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 1.87e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 5.73e8T + 5.11e17T^{2} \) |
| 61 | \( 1 - 4.87e7iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 6.11e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.16e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 5.90e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 3.70e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 4.86e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 6.61e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 3.91e9T + 7.37e19T^{2} \) |
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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
−14.50991076253474179346929529503, −12.61928970884649689127718323652, −11.69596372447519500179604452501, −10.39743034805960630842340611771, −9.225309340828950779814408030237, −8.098121369228500701752421463293, −6.08002166719105451341338815040, −4.99866959931267792570343872633, −3.41335940717006964194139015941, −1.57157701752995067998164040924,
0.877861647262700336472932672793, 2.11924837575118649406163093076, 3.52722088933423811893283740617, 6.24922391192616224171945796676, 7.01573305834523971647291821666, 7.83771102526070773744104381345, 10.17405398034777853776165082793, 11.11310896409288890889673808593, 12.15505657397960373074973713902, 13.41387254006604936600466853223
