L(s) = 1 | + (0.866 + 0.5i)3-s + (0.218 + 0.816i)7-s + (0.499 + 0.866i)9-s + (−1.15 + 0.811i)13-s + (−0.524 − 1.43i)19-s + (−0.218 + 0.816i)21-s + (0.642 − 0.766i)25-s + 0.999i·27-s + (−0.142 − 1.63i)31-s + (1.62 + 0.592i)37-s + (−1.40 + 0.123i)39-s + (−0.168 + 0.0451i)43-s + (0.247 − 0.142i)49-s + (0.266 − 1.50i)57-s + (−1.93 + 0.342i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.218 + 0.816i)7-s + (0.499 + 0.866i)9-s + (−1.15 + 0.811i)13-s + (−0.524 − 1.43i)19-s + (−0.218 + 0.816i)21-s + (0.642 − 0.766i)25-s + 0.999i·27-s + (−0.142 − 1.63i)31-s + (1.62 + 0.592i)37-s + (−1.40 + 0.123i)39-s + (−0.168 + 0.0451i)43-s + (0.247 − 0.142i)49-s + (0.266 − 1.50i)57-s + (−1.93 + 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.284052998\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284052998\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
good | 5 | \( 1 + (-0.642 + 0.766i)T^{2} \) |
| 7 | \( 1 + (-0.218 - 0.816i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.811i)T + (0.342 - 0.939i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.524 + 1.43i)T + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 31 | \( 1 + (0.142 + 1.63i)T + (-0.984 + 0.173i)T^{2} \) |
| 37 | \( 1 + (-1.62 - 0.592i)T + (0.766 + 0.642i)T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.168 - 0.0451i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.342 + 0.939i)T^{2} \) |
| 53 | \( 1 + (0.642 + 0.766i)T^{2} \) |
| 59 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 61 | \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.342 + 0.0603i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \) |
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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
−10.30307716254044580651617073400, −9.369251982166857635972015424115, −9.017345019454203025108806052175, −8.046940666779029999418297720791, −7.23382900168659013278366173587, −6.15439318781586863626831710678, −4.83633821205297865438840438200, −4.36153357765731827025514669982, −2.81057924664315450848766762383, −2.19364054274912134156601082890,
1.39011864945304327806498006748, 2.73592855306068369472590667778, 3.72334611509445472983828971752, 4.77414850729590462360641319655, 6.01195323799810015195109739956, 7.14160735526857161757082313275, 7.64160108998842040475694310020, 8.405191715233195137108321915169, 9.381379160494905756985466830569, 10.18903602611792987853373704448