L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.707 + 0.707i)20-s + (0.866 − 0.499i)25-s − 1.93i·26-s + (0.258 − 0.448i)29-s + (0.258 − 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.707 + 0.707i)20-s + (0.866 − 0.499i)25-s − 1.93i·26-s + (0.258 − 0.448i)29-s + (0.258 − 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.822443214\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822443214\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.965 - 0.258i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 41 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.93T + T^{2} \) |
| 97 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.886570933456060098006701164930, −8.267529327029634857449384159745, −7.987136880002714804208652749874, −7.01055489132790251654518729923, −6.04519781778741095813716057039, −5.40200484753564014770791763470, −4.35164164595997464062830048294, −3.43723002873263752008228732677, −2.95608420041304829326691641101, −1.25719470074902281596469105692,
1.66228475433955525174610720029, 3.10632177415839527912161308970, 3.84694304241073302815922391237, 4.69084544375554521517513565773, 5.37530661828103852456376742454, 6.60276728275919689309488735449, 7.08929890066321207634765751633, 7.914571285032784022255580883283, 8.721379209449021083528438756253, 9.577299949292323914621764842667