| L(s) = 1 | + (0.0532 − 0.198i)2-s + (−2.75 − 4.76i)3-s + (3.42 + 1.97i)4-s + (−1.09 + 0.292i)6-s + (1.16 + 4.33i)7-s + (1.15 − 1.15i)8-s + (−10.6 + 18.4i)9-s + (−18.8 − 5.05i)11-s − 21.7i·12-s + (−8.04 − 10.2i)13-s + 0.923·14-s + (7.74 + 13.4i)16-s + (−10.6 − 6.15i)17-s + (3.09 + 3.09i)18-s + (−11.7 + 3.15i)19-s + ⋯ |
| L(s) = 1 | + (0.0266 − 0.0993i)2-s + (−0.916 − 1.58i)3-s + (0.856 + 0.494i)4-s + (−0.182 + 0.0488i)6-s + (0.165 + 0.619i)7-s + (0.144 − 0.144i)8-s + (−1.18 + 2.04i)9-s + (−1.71 − 0.459i)11-s − 1.81i·12-s + (−0.619 − 0.785i)13-s + 0.0659·14-s + (0.484 + 0.838i)16-s + (−0.627 − 0.362i)17-s + (0.171 + 0.171i)18-s + (−0.619 + 0.165i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0387320 + 0.0839379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0387320 + 0.0839379i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (8.04 + 10.2i)T \) |
| good | 2 | \( 1 + (-0.0532 + 0.198i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (2.75 + 4.76i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (-1.16 - 4.33i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (18.8 + 5.05i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (10.6 + 6.15i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (11.7 - 3.15i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (28.0 - 16.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (0.962 + 1.66i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-25.6 - 25.6i)T + 961iT^{2} \) |
| 37 | \( 1 + (26.8 + 7.20i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 38.7i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (0.993 + 0.573i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (31.6 - 31.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 26.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (1.92 + 7.16i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-21.1 + 36.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.01 - 7.53i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-26.8 + 7.19i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-77.2 + 77.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 77.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (7.65 + 7.65i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (149. + 40.1i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (141. - 37.9i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−11.04026913006097933007292850463, −10.36840979429208911274582999240, −8.291415245995119384954056511442, −7.83982832604670750195065485466, −6.94673951896502578465514518734, −5.94571408515582459130445743130, −5.21648231196178809985732008727, −2.79159003864142659436484168860, −1.98733408586560751663142695855, −0.04232239336771460278720027544,
2.46319603445626659460547014034, 4.23089674308609529239715239191, 4.93490395753108470738838730038, 5.95351689162337015465125299099, 6.91411661391292296418118221650, 8.230297615638795297079406728355, 9.798312046726908994347075782050, 10.20049471529622759424643358282, 10.88971665987789065698931600800, 11.56083200665958988726894552921
