L(s) = 1 | + (−0.866 + 0.5i)3-s + (−0.5 − 1.86i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)13-s + (1.36 − 0.366i)19-s + (1.36 + 1.36i)21-s − i·25-s + 0.999i·27-s + (0.366 + 0.366i)31-s + (0.366 − 1.36i)37-s + 0.999·39-s + (−0.5 + 0.866i)43-s + (−2.36 + 1.36i)49-s + (−0.999 + i)57-s + (−0.866 + 1.5i)61-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (−0.5 − 1.86i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)13-s + (1.36 − 0.366i)19-s + (1.36 + 1.36i)21-s − i·25-s + 0.999i·27-s + (0.366 + 0.366i)31-s + (0.366 − 1.36i)37-s + 0.999·39-s + (−0.5 + 0.866i)43-s + (−2.36 + 1.36i)49-s + (−0.999 + i)57-s + (−0.866 + 1.5i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6108614497\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6108614497\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 37 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 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
−10.50166965958906157216421912121, −10.06149869451287955972074173062, −9.312780598764084002743101039789, −7.71070863978518065064309355484, −7.12178598663380236040742671321, −6.20307338343528435494364435340, −5.01352425793039458520971770938, −4.22009431994373522937465165834, −3.18693731212803509935053836816, −0.805185622971814295299179484791,
1.89311658766479411981328128539, 3.07251215638947684371872781925, 4.89395072277040478142646555029, 5.55192510052919265858584244269, 6.36457961974624249878133078678, 7.29989393595632467944713290454, 8.315205720545005785757717558074, 9.406752352808669447380759493125, 9.936398568455412265000168408052, 11.32803521643644268022226661697