L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.448i)5-s + (0.258 − 0.965i)6-s − 0.999i·8-s − 1.00i·9-s − 0.517i·10-s + (−0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 1.93i·19-s + (−0.258 − 0.448i)20-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.448i)5-s + (0.258 − 0.965i)6-s − 0.999i·8-s − 1.00i·9-s − 0.517i·10-s + (−0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 1.93i·19-s + (−0.258 − 0.448i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.883203160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.883203160\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.93iT - T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.539287753706819535234253478011, −7.80791352098717819694396502803, −6.89032356900973158370782244184, −6.02307553965640846598594134922, −5.79625494132835012097172012986, −4.36580316403611913493660313134, −3.85917576982379725746295656181, −2.98162981875803574789574816959, −1.83754609494224675549900643735, −1.36241478810149651874768792621,
2.00421231405516646786522089724, 2.93617984704579203909392690031, 3.50285949975553282592395804252, 4.37978244184015210288041785365, 5.09694961212281206803148217746, 5.96548328608091900784780182081, 6.59570615369864306793185557844, 7.56384219315667568411090331512, 8.130315450023434524427952209503, 8.843658865022076793192077802637