L(s) = 1 | + i·4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)13-s − 16-s + (0.5 + 0.133i)19-s + (0.866 − 0.5i)25-s + (−0.5 + 0.866i)28-s + (−1.36 − 0.366i)31-s + (0.366 + 0.366i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s − i·64-s + (−0.366 − 1.36i)67-s + ⋯ |
L(s) = 1 | + i·4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)13-s − 16-s + (0.5 + 0.133i)19-s + (0.866 − 0.5i)25-s + (−0.5 + 0.866i)28-s + (−1.36 − 0.366i)31-s + (0.366 + 0.366i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s − i·64-s + (−0.366 − 1.36i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.018087575\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.018087575\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−10.81221757945136576433777930049, −9.474683452929971466347792091887, −8.920243341890646112675628599459, −7.895346977098269944492767465347, −7.44263098902453268022039894309, −6.32081880523776641423373293890, −5.06868123916336621326220287199, −4.33702109191745518472809897415, −3.07170183736791462023445336777, −2.01939365502429053450448712524,
1.18537092265828073204698414618, 2.51422642931971795856576461271, 4.09155526514782354611348265603, 5.11254612560912582179466153225, 5.64262534157675612940134875579, 7.00775148587544687291411407312, 7.53716428980118435737982812071, 8.748658300376336932787269998767, 9.507256784530401311334466114653, 10.45115370678247060667368600669