L(s) = 1 | − 3-s + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s + 9-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.866 + 0.5i)15-s + (−0.866 − 0.5i)21-s + (−0.866 + 0.5i)23-s − 27-s + (−0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + 0.999·35-s + (−0.866 + 0.5i)39-s + ⋯ |
L(s) = 1 | − 3-s + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s + 9-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.866 + 0.5i)15-s + (−0.866 − 0.5i)21-s + (−0.866 + 0.5i)23-s − 27-s + (−0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + 0.999·35-s + (−0.866 + 0.5i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.192681572\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192681572\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.250619835087223170447446399093, −8.323965905397429425951764585002, −7.74856710574565273406541674996, −6.39506427588180373013899091152, −5.95377329283669784835266598038, −5.38090905584293474919846386238, −4.56054713891804340614846476215, −3.52058008210375913596753546061, −1.96865519969110839396338875293, −1.13902798561168856955780780498,
1.39376195945581395235393614584, 2.11204631711695120465558359751, 3.80326017057811259249567934173, 4.51343738225797053942682012757, 5.34892765313802185912161046450, 6.24183856855891407736652194191, 6.72646694174215307741801291706, 7.50181288602220005503120415109, 8.459626056790001875929797463976, 9.494082850667994758599063058737