L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999·6-s + (0.366 − 1.36i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (−0.866 − 0.499i)12-s + (0.866 − 0.5i)13-s + (−1 + 0.999i)14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.499i)20-s + (0.366 + 1.36i)21-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999·6-s + (0.366 − 1.36i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (−0.866 − 0.499i)12-s + (0.866 − 0.5i)13-s + (−1 + 0.999i)14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.499i)20-s + (0.366 + 1.36i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7849961108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7849961108\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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
−8.618025208900398850906023694596, −8.452999639850065202727594773213, −7.20224763848325864685107427029, −6.49243069843905990273775438129, −5.76883465240412059866170189786, −4.78170971433015563940957216927, −3.98535081963709606700404993358, −3.07896274547623342882684593816, −1.53891220872824419082944388300, −0.802777985327476329299920031382,
1.59342331609734492688552362480, 1.88560717501294753247164714167, 3.34731282323365475947121039591, 5.06760941968196732492695794686, 5.52920115232650616021372752894, 6.25899944980471422820934320094, 6.78126187168490535282419685989, 7.32214284153172217799725399058, 8.623736167333531574001745865788, 9.088768288201378821917971293003