L(s) = 1 | + 3-s + (−0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + i·21-s + (0.866 + 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)35-s + ⋯ |
L(s) = 1 | + 3-s + (−0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + i·21-s + (0.866 + 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.497729248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.497729248\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−9.217246590838677211255777616815, −8.322088411619331535590297798613, −8.062788948886037319567431432696, −7.31577227740803053383002885185, −5.99238593005154809693645393088, −5.33687866218894042104795589000, −4.23622190833605931192761261269, −3.32490906733874873109772551917, −2.72685212751734310627748068794, −1.21600198531380253140492633867,
1.39083920379539830629119355680, 2.69039443357385505142770334513, 3.70051467709518397146510503186, 4.08589529023136171634107614193, 5.14163316858762065888393606682, 6.64867806659210168536982905695, 7.20933111378829073322847311802, 7.71032485523505100559368044197, 8.499319340976909184680925155318, 9.298979959879874296688582488868