| L(s) = 1 | + 11.0i·2-s + 50.0i·3-s − 57.2·4-s − 30.3i·5-s − 551.·6-s + 75.1i·7-s + 74.1i·8-s − 1.77e3·9-s + 333.·10-s + 604.·11-s − 2.86e3i·12-s + 2.63e3·13-s − 827.·14-s + 1.51e3·15-s − 4.48e3·16-s + 4.98e3·17-s + ⋯ |
| L(s) = 1 | + 1.37i·2-s + 1.85i·3-s − 0.894·4-s − 0.242i·5-s − 2.55·6-s + 0.219i·7-s + 0.144i·8-s − 2.43·9-s + 0.333·10-s + 0.454·11-s − 1.65i·12-s + 1.19·13-s − 0.301·14-s + 0.449·15-s − 1.09·16-s + 1.01·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.852844 - 1.26927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.852844 - 1.26927i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.00e4 + 7.36e4i)T \) |
| good | 2 | \( 1 - 11.0iT - 64T^{2} \) |
| 3 | \( 1 - 50.0iT - 729T^{2} \) |
| 5 | \( 1 + 30.3iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 75.1iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 604.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.63e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.98e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 3.03e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.24e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.44e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.93e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.30e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.34e4T + 4.75e9T^{2} \) |
| 47 | \( 1 + 1.57e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.12e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.63e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 1.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.57e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.85e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.85e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 7.62e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 2.68e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.32e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.43e6T + 8.32e11T^{2} \) |
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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
−15.59124975924227519397348915297, −14.79780476998469971375374729811, −13.90912453237388092168942836563, −11.69593927701318749390995176081, −10.39207399791418413615774567024, −9.107001507201191564784400154912, −8.229856241815648976330009124998, −6.12937542079977942229898723948, −5.11118008594230860008750925722, −3.70050677010692192441423647576,
0.75803919748514223617647192878, 1.84622879162317828705514349604, 3.35705503837447035781849292371, 6.17449157945888580298653907289, 7.42239510999107766999968725780, 8.910242601321135745049043590212, 10.72826587238598475614308894133, 11.69355018631850048319353974735, 12.56397430847526334818805625696, 13.41842365017312428017187276904
