| L(s) = 1 | + (0.654 − 0.755i)3-s + (−0.192 + 0.123i)5-s + (−3.39 − 1.36i)7-s + (−0.142 − 0.989i)9-s + (0.270 + 5.68i)11-s + (−7.01 + 0.669i)13-s + (−0.0325 + 0.226i)15-s + (−0.521 − 2.15i)17-s + (3.17 − 1.27i)19-s + (−3.25 + 1.67i)21-s + (−5.32 − 1.02i)23-s + (−2.05 + 4.50i)25-s + (−0.841 − 0.540i)27-s + (−2.03 + 3.53i)29-s + (−4.18 − 0.399i)31-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.436i)3-s + (−0.0859 + 0.0552i)5-s + (−1.28 − 0.514i)7-s + (−0.0474 − 0.329i)9-s + (0.0816 + 1.71i)11-s + (−1.94 + 0.185i)13-s + (−0.00839 + 0.0584i)15-s + (−0.126 − 0.521i)17-s + (0.727 − 0.291i)19-s + (−0.710 + 0.366i)21-s + (−1.11 − 0.213i)23-s + (−0.411 + 0.900i)25-s + (−0.161 − 0.104i)27-s + (−0.378 + 0.655i)29-s + (−0.752 − 0.0718i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0534220 + 0.178774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0534220 + 0.178774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (4.45 - 6.86i)T \) |
| good | 5 | \( 1 + (0.192 - 0.123i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (3.39 + 1.36i)T + (5.06 + 4.83i)T^{2} \) |
| 11 | \( 1 + (-0.270 - 5.68i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (7.01 - 0.669i)T + (12.7 - 2.46i)T^{2} \) |
| 17 | \( 1 + (0.521 + 2.15i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (-3.17 + 1.27i)T + (13.7 - 13.1i)T^{2} \) |
| 23 | \( 1 + (5.32 + 1.02i)T + (21.3 + 8.54i)T^{2} \) |
| 29 | \( 1 + (2.03 - 3.53i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.18 + 0.399i)T + (30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-5.38 - 9.32i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.45 + 6.15i)T + (1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.315 + 0.0925i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (3.08 + 8.90i)T + (-36.9 + 29.0i)T^{2} \) |
| 53 | \( 1 + (4.04 - 1.18i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.67 + 10.2i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.371 - 7.80i)T + (-60.7 - 5.79i)T^{2} \) |
| 71 | \( 1 + (-0.0517 + 0.213i)T + (-63.1 - 32.5i)T^{2} \) |
| 73 | \( 1 + (-0.400 + 8.40i)T + (-72.6 - 6.93i)T^{2} \) |
| 79 | \( 1 + (0.176 + 0.247i)T + (-25.8 + 74.6i)T^{2} \) |
| 83 | \( 1 + (11.5 + 5.97i)T + (48.1 + 67.6i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 12.4i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (5.87 + 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.22830122957195867535877738196, −9.665742685107065122647058010942, −9.283013748810273092835175268948, −7.61428798195150321938678699184, −7.28393928659980405979525475860, −6.60763830664744432973286326394, −5.20657567113097580618497599150, −4.20880467448493225837513510413, −3.02343089359884643580068091064, −1.99413632882952046518166041501,
0.080218801711612889845330320619, 2.47494417574051321889133375700, 3.26922062056770215689091245129, 4.29060589506214194125485133620, 5.69690342524916400762772070187, 6.13276667325160897601572384300, 7.53700619665376413428336738011, 8.200388970911160826562806140091, 9.493494319254301619737248304114, 9.521157647691481954877023851890
