L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 2·19-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 2·19-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04353077375\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04353077375\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + 2T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−8.967132347421788892073667049175, −8.534624625904106299465286447631, −7.929555447329775812759920320191, −6.76816349997819403265102159720, −6.24585507867256146063172032112, −5.90109163867976834950857776994, −4.71928605413644973607799073780, −4.07189271107812219881674394447, −2.76232506792537979809814918890, −1.72288800520640974622082928013,
0.02842847575095271850409578002, 1.65891832304948384759162438358, 2.37502415471120635942855888509, 3.66841486213620620685021198556, 4.12876320244910594769997603844, 4.86538076322577173850613884241, 6.27008775833454374760507993378, 6.95609358637747146699893211448, 7.52250402813517444638418819208, 8.639279730557338929301860622533