L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.965 + 0.258i)5-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s − i·10-s + (0.258 + 0.965i)11-s + (1.36 + 0.366i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + 1.41i·17-s + (−0.965 + 0.258i)20-s + (0.866 − 0.499i)22-s − 1.41i·26-s + 28-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.965 + 0.258i)5-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s − i·10-s + (0.258 + 0.965i)11-s + (1.36 + 0.366i)13-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + 1.41i·17-s + (−0.965 + 0.258i)20-s + (0.866 − 0.499i)22-s − 1.41i·26-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9882474937\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9882474937\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.808730740595481900235331303961, −9.296097027963065656996273511724, −8.450944681937003754591213411874, −7.38344457780492919948586911401, −6.38310593739596787665124735952, −5.70113740837317692432495863931, −4.20086017386142825253141147455, −3.70209898163036016745239710555, −2.38161954190077160799018605216, −1.45814919120194475032648610554,
1.11420103488335637016578567678, 2.85975602498092472538616548569, 3.97472684557426941376840535016, 5.40175348406785170707641278323, 5.78565247162744099724032908379, 6.47729429014089947964380918644, 7.37193173208057527433763425955, 8.548626345454152969343817295313, 9.027432690034710120313973890764, 9.577380895189716509329930671092