L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (−0.173 + 0.300i)5-s + (−0.939 + 0.342i)6-s + (0.939 + 1.62i)7-s + 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.939 − 1.62i)14-s + (0.266 + 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (0.173 + 0.984i)18-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (−0.173 + 0.300i)5-s + (−0.939 + 0.342i)6-s + (0.939 + 1.62i)7-s + 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.939 − 1.62i)14-s + (0.266 + 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (0.173 + 0.984i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8297608260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8297608260\) |
\(L(1)\) |
|
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 | \( 1 + (-0.173 + 0.984i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 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 | \( 1 + 1.87T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.12561953246003034088702964194, −9.177407836127679575888622890095, −8.566600047167952711145904534522, −7.85110253195227991390778607808, −7.10858294882187443356047848275, −5.81080502674270155156124744950, −4.91400903584367912686797286196, −3.36204916651749334758280674228, −2.40047327133074093986645386962, −1.59866572645191198318417654364,
1.10026601428616480244844075527, 3.30957979132894864975853420282, 4.55715722378081075734004830951, 4.87686120309589469983735011763, 5.99259081473182796539335119032, 7.29332531463778261290621667309, 7.959531437037656939703118003542, 8.414796891849328827458901515844, 9.668222109469962129174184650609, 10.22074151585986818239991184121