L(s) = 1 | + 2-s + 4-s + 8-s + 16-s + (−1 − i)17-s + (1 − i)19-s + (1 + i)23-s + 32-s + (−1 − i)34-s + (1 − i)38-s + (1 + i)46-s + (−1 − i)47-s + i·49-s + 2i·53-s + (1 + i)61-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 8-s + 16-s + (−1 − i)17-s + (1 − i)19-s + (1 + i)23-s + 32-s + (−1 − i)34-s + (1 − i)38-s + (1 + i)46-s + (−1 − i)47-s + i·49-s + 2i·53-s + (1 + i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.559874638\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.559874638\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.790992238917499576246919786578, −7.57513745641619228211733496290, −7.18856308503068707964419483434, −6.47241231574239751317512153207, −5.50470912544726460002159708056, −4.94797969430622192602973608200, −4.20836516324813963644486868443, −3.13808068165915260415013603914, −2.58104332158666329411768478999, −1.28524364473144289524865885834,
1.44904393535381829465750433600, 2.43928531236859538920031795290, 3.40537873563978453697267780075, 4.11156005976907006916252028381, 4.97765505645303978347063780143, 5.63142473548678773609038152371, 6.55904275422440354084720007923, 6.93020166853298988510859527003, 8.026537898858517693773409798944, 8.478989919133056169283558066796