L(s) = 1 | + 2.82i·2-s + 5.19·3-s − 8.00·4-s − 4.63·5-s + 14.6i·6-s − 33.3·7-s − 22.6i·8-s + 27·9-s − 13.1i·10-s − 4.16i·11-s − 41.5·12-s + 241. i·13-s − 94.2i·14-s − 24.0·15-s + 64.0·16-s + 291.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.185·5-s + 0.408i·6-s − 0.680·7-s − 0.353i·8-s + 0.333·9-s − 0.131i·10-s − 0.0344i·11-s − 0.288·12-s + 1.43i·13-s − 0.480i·14-s − 0.107·15-s + 0.250·16-s + 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3628064280\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3628064280\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (3.41e3 + 667. i)T \) |
good | 5 | \( 1 + 4.63T + 625T^{2} \) |
| 7 | \( 1 + 33.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 4.16iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 241. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 291.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 369.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 496. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 329.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.48e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 727. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.74e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.68e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.18e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.22e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.20e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 279. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.00e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.62e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 2.78e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.08e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.40e4iT - 8.85e7T^{2} \) |
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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
−10.27106178561596689578695314793, −9.463735930741157225977616599895, −8.674996157098224389901107434750, −7.71958513547454524218685067878, −6.78104676572686352134196783489, −5.93370471565183078280192421793, −4.45085526902142947087988471112, −3.59170661644286481965459003545, −2.03368965589564736000226829042, −0.096099197597186156604388816813,
1.45222008787349342426416786434, 2.98765756408880730767966043889, 3.58563547573110218317587871935, 5.02090224798467708258759941723, 6.21210241153954546670666788605, 7.63382948592119311880911151228, 8.323128549643662665120491139310, 9.480434320018311506520357186167, 10.11424458512633432560036005156, 10.96446084593624909302707239441