L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 − 0.5i)10-s + 0.999i·12-s − 0.999·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 − 0.5i)10-s + 0.999i·12-s − 0.999·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804378035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804378035\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + 2iT - T^{2} \) |
| 29 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−10.12138138219703362379779919558, −9.021106584700497471244897484133, −8.396080466477199243907773762826, −7.74017106697005777442533150196, −6.80200915222145220340710965467, −6.02831129149565488126926598592, −4.65159310679076219922108652710, −4.24942676342106737749656615011, −3.10502512536619287865005350163, −2.60121535867422162821335173420,
1.18522564287751680934537943335, 2.70895887295229795491606934733, 3.52041502107327942795861360787, 4.03295640683285972302453678291, 5.25932466250393343784622561448, 6.52896801410802086717922928816, 7.08306263283818787325872646738, 7.83402656602402720356312652080, 8.906306890734304115196595285779, 9.722903308503074943187783227164