L(s) = 1 | − 31.2i·2-s − 113. i·3-s − 722.·4-s + 959. i·5-s − 3.55e3·6-s − 3.35e3i·7-s + 1.45e4i·8-s − 6.36e3·9-s + 3.00e4·10-s + 5.04e3·11-s + 8.21e4i·12-s − 1.34e4·13-s − 1.04e5·14-s + 1.09e5·15-s + 2.71e5·16-s − 9.36e4·17-s + ⋯ |
L(s) = 1 | − 1.95i·2-s − 1.40i·3-s − 2.82·4-s + 1.53i·5-s − 2.74·6-s − 1.39i·7-s + 3.55i·8-s − 0.969·9-s + 3.00·10-s + 0.344·11-s + 3.95i·12-s − 0.471·13-s − 2.73·14-s + 2.15·15-s + 4.13·16-s − 1.12·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0410i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 + 0.0410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0792654 - 0.00162917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0792654 - 0.00162917i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-3.41e6 - 1.40e5i)T \) |
good | 2 | \( 1 + 31.2iT - 256T^{2} \) |
| 3 | \( 1 + 113. iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 959. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.35e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 5.04e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + 1.34e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 9.36e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 7.49e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 3.09e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 5.68e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 9.53e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.56e3iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.64e5T + 7.98e12T^{2} \) |
| 47 | \( 1 + 8.56e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 3.81e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + 9.85e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.09e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.57e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.72e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.85e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.99e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.39e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.45e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.45e7T + 7.83e15T^{2} \) |
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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
−12.87171568847024571610958129099, −11.71596467043836512483885173626, −10.81291208768388242222012363881, −9.937990498851729068332933826263, −7.950756034853201879990048587811, −6.67768101161337376491709214198, −4.02640561826675823896748544893, −2.65924719454184162292879364774, −1.47419040653978984525823255733, −0.03167890074263516998636272734,
4.35567461122193415654533454167, 4.98820545894514746495095162002, 6.07925253410874916244434798741, 8.248736824636466069034490251199, 9.055836903432271076011325182292, 9.615372302014921552518256081308, 12.20128632538444716436211834228, 13.46145759237487828111259906272, 14.89287626130262453286875976413, 15.63178101755690426442820680749