L(s) = 1 | + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)11-s + (0.707 − 0.707i)13-s − 15-s + (0.258 − 0.965i)17-s + (−1.22 − 0.707i)19-s + (0.366 + 1.36i)23-s + (0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − i·29-s + (−0.258 − 0.965i)33-s + (1.36 − 0.366i)37-s + (0.866 − 0.500i)39-s + (1 − i)43-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)11-s + (0.707 − 0.707i)13-s − 15-s + (0.258 − 0.965i)17-s + (−1.22 − 0.707i)19-s + (0.366 + 1.36i)23-s + (0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − i·29-s + (−0.258 − 0.965i)33-s + (1.36 − 0.366i)37-s + (0.866 − 0.500i)39-s + (1 − i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.242652337\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242652337\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
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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
−8.506697637138671975886575771458, −7.85715817237633165833341730630, −7.40109401913310469631733243331, −6.29547640807734331795262316313, −5.57617281295906960235844836516, −4.51151864009146424748954003763, −3.70247410072498112661578133924, −3.09595531842012476603387382358, −2.44965295178902116507691057748, −0.62641060186266334874102571313,
1.48481874723939200215319785259, 2.41705656978140275097639512051, 3.33513936163986294472662268923, 4.18800389417157308571876606194, 4.69253541618586048570368977219, 5.93561449524976088031502893224, 6.72918852421487723920794864739, 7.51232411728634712894758758787, 8.260097629353807957447119034535, 8.464381885031253070965494939183