L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)11-s − 0.999i·15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 − 0.5i)23-s − i·27-s + (0.866 + 0.5i)31-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)47-s + (0.866 − 0.499i)51-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)11-s − 0.999i·15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 − 0.5i)23-s − i·27-s + (0.866 + 0.5i)31-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)47-s + (0.866 − 0.499i)51-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.592568554\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592568554\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.769611623537375564558293126678, −8.398603515250054868804541143514, −7.44253374519593925678681622326, −6.69619959454137975511671819996, −5.69078379908374637099113452688, −4.63150322974603807271831795485, −4.20379713348296201976610130464, −3.33797675643618457444676892905, −2.38447939447465937596117041548, −1.00950127237202719727526351339,
1.39330557589104093864253776545, 2.50857466775479624239295860558, 3.28105804605966295971960358963, 3.90198232208884329957194387110, 5.04820145270392473528533659549, 6.17426141720240470605571973949, 6.78369857633977121317662359171, 7.45965324600067064770947780007, 8.213764015173753110777138946209, 8.687070477881930723300090990620