L(s) = 1 | + (0.388 − 1.35i)2-s + (−1.69 − 1.05i)4-s − 1.84i·5-s + 2.17i·7-s + (−2.09 + 1.89i)8-s + (−2.50 − 0.716i)10-s + 4.96·11-s + 3.69·13-s + (2.96 + 0.846i)14-s + (1.76 + 3.58i)16-s + 2.85·17-s + (−3.47 + 2.63i)19-s + (−1.94 + 3.13i)20-s + (1.92 − 6.75i)22-s − 7.16i·23-s + ⋯ |
L(s) = 1 | + (0.274 − 0.961i)2-s + (−0.849 − 0.528i)4-s − 0.824i·5-s + 0.823i·7-s + (−0.740 + 0.671i)8-s + (−0.793 − 0.226i)10-s + 1.49·11-s + 1.02·13-s + (0.792 + 0.226i)14-s + (0.442 + 0.896i)16-s + 0.692·17-s + (−0.796 + 0.604i)19-s + (−0.435 + 0.700i)20-s + (0.411 − 1.43i)22-s − 1.49i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.958225619\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958225619\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.388 + 1.35i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.47 - 2.63i)T \) |
good | 5 | \( 1 + 1.84iT - 5T^{2} \) |
| 7 | \( 1 - 2.17iT - 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 23 | \( 1 + 7.16iT - 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 + 0.613iT - 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 7.16iT - 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.64iT - 59T^{2} \) |
| 61 | \( 1 + 1.26iT - 61T^{2} \) |
| 67 | \( 1 + 3.30iT - 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 3.23iT - 89T^{2} \) |
| 97 | \( 1 - 4.47iT - 97T^{2} \) |
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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
−9.282420576908010942000908543075, −8.658378753274674866679699917529, −8.364969118905188139471995055107, −6.55078970433986118170369427477, −5.95519604464040737256480795083, −4.93049428552256127746734076792, −4.15275322893725049404878928970, −3.24950543573570273838548130546, −1.92848444004173844012521086889, −0.959449851266569723925943098151,
1.19360740520169155418989909669, 3.23018233833255353604152291324, 3.81249068196591353141953577710, 4.73575165548763415381262303110, 5.99021945450458731357014187035, 6.60162257199304483927780240393, 7.14507906698043755174570574452, 8.036055131928566223479231304577, 8.915944383993592074203614377306, 9.618479883756977969369211848121