| L(s) = 1 | − 16.6i·2-s + 405. i·3-s + 745.·4-s − 2.50e3i·5-s + 6.76e3·6-s − 1.11e4i·7-s − 2.95e4i·8-s − 1.05e5·9-s − 4.18e4·10-s + 7.68e4·11-s + 3.02e5i·12-s − 6.13e5·13-s − 1.85e5·14-s + 1.01e6·15-s + 2.71e5·16-s − 2.69e6·17-s + ⋯ |
| L(s) = 1 | − 0.521i·2-s + 1.67i·3-s + 0.728·4-s − 0.802i·5-s + 0.870·6-s − 0.661i·7-s − 0.900i·8-s − 1.78·9-s − 0.418·10-s + 0.477·11-s + 1.21i·12-s − 1.65·13-s − 0.344·14-s + 1.34·15-s + 0.258·16-s − 1.89·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.629743 - 0.972809i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629743 - 0.972809i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.01e7 + 1.34e8i)T \) |
| good | 2 | \( 1 + 16.6iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 405. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 2.50e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 1.11e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 7.68e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + 6.13e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.69e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 2.92e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 2.15e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 3.40e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 3.94e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 1.06e8iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 6.32e7T + 1.34e16T^{2} \) |
| 47 | \( 1 + 7.95e7T + 5.25e16T^{2} \) |
| 53 | \( 1 - 5.81e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 6.19e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + 7.79e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 + 6.11e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 7.76e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.33e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 2.30e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 9.15e8T + 1.55e19T^{2} \) |
| 89 | \( 1 - 4.73e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.01e10T + 7.37e19T^{2} \) |
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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
−13.27556947151011994784275830664, −11.79590691222134619443667741433, −10.88312696392129570406557641701, −9.859997214919649024208786069058, −8.957501409971747860282887994316, −6.94747353903272324255367912553, −4.93784891787546814001778554297, −4.06418995302411805696953069564, −2.45854724755955475173281688251, −0.32949994621304945916876958639,
1.85450405856959863719303261701, 2.63856674118069847145061674294, 5.67119789454148018791768177608, 6.91230860262212842913998476207, 7.24459957621074172686842197913, 8.747198155167312444893675871131, 10.89719104193704380455090811209, 11.96450234259281955029062251214, 12.78643392883178107857674965163, 14.42578624949313680506173265291
