L(s) = 1 | + (0.870 − 0.491i)2-s + (−0.309 + 0.951i)3-s + (0.516 − 0.856i)4-s + (−0.941 + 0.336i)5-s + (0.198 + 0.980i)6-s + (0.0285 − 0.999i)8-s + (−0.809 − 0.587i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (0.0855 + 0.996i)13-s + (−0.0285 − 0.999i)15-s + (−0.466 − 0.884i)16-s + (−0.998 + 0.0570i)17-s + (−0.993 − 0.113i)18-s + (−0.254 − 0.967i)19-s + (−0.198 + 0.980i)20-s + ⋯ |
L(s) = 1 | + (0.870 − 0.491i)2-s + (−0.309 + 0.951i)3-s + (0.516 − 0.856i)4-s + (−0.941 + 0.336i)5-s + (0.198 + 0.980i)6-s + (0.0285 − 0.999i)8-s + (−0.809 − 0.587i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (0.0855 + 0.996i)13-s + (−0.0285 − 0.999i)15-s + (−0.466 − 0.884i)16-s + (−0.998 + 0.0570i)17-s + (−0.993 − 0.113i)18-s + (−0.254 − 0.967i)19-s + (−0.198 + 0.980i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03302063792 - 0.2234814857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03302063792 - 0.2234814857i\) |
\(L(1)\) |
\(\approx\) |
\(0.9554650422 - 0.06007164400i\) |
\(L(1)\) |
\(\approx\) |
\(0.9554650422 - 0.06007164400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.870 - 0.491i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.941 + 0.336i)T \) |
| 13 | \( 1 + (0.0855 + 0.996i)T \) |
| 17 | \( 1 + (-0.998 + 0.0570i)T \) |
| 19 | \( 1 + (-0.254 - 0.967i)T \) |
| 23 | \( 1 + (-0.142 + 0.989i)T \) |
| 29 | \( 1 + (-0.774 - 0.633i)T \) |
| 31 | \( 1 + (-0.974 + 0.226i)T \) |
| 37 | \( 1 + (-0.921 - 0.389i)T \) |
| 41 | \( 1 + (-0.736 - 0.676i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 + (-0.993 + 0.113i)T \) |
| 53 | \( 1 + (-0.466 + 0.884i)T \) |
| 59 | \( 1 + (0.736 - 0.676i)T \) |
| 61 | \( 1 + (-0.870 - 0.491i)T \) |
| 67 | \( 1 + (0.415 - 0.909i)T \) |
| 71 | \( 1 + (-0.362 + 0.931i)T \) |
| 73 | \( 1 + (-0.985 + 0.170i)T \) |
| 79 | \( 1 + (-0.610 - 0.791i)T \) |
| 83 | \( 1 + (0.696 + 0.717i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.941 - 0.336i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.6428102403260114081320715788, −22.15726185373215335470763087150, −20.685400061570894772937076850271, −20.23532445044557299633560368850, −19.390642659060851450230310291298, −18.37493969597559392554148576821, −17.61754944947301747278434802811, −16.63134109898183790965262563119, −16.15296417262570440433645350801, −15.06890521518543572345988223899, −14.504063971318874286735328035078, −13.311401017261613369165120749109, −12.80366575218494750422849647117, −12.17848657053389702910025566430, −11.343240179128857815278958206, −10.626043181757357360436552692383, −8.72240031520519113807919193839, −8.1430880801935566202258152253, −7.37854023725225038062005356909, −6.603256990439348543641018411209, −5.66714156455165151115970220916, −4.84723152840582825182119400356, −3.80267620343647884140515854734, −2.83952880318940121587236439098, −1.63099507773758086002153106767,
0.07283971088527305270680609599, 1.94529772974658178131384180164, 3.13215699744467921500466210603, 3.94719673901922843381509614606, 4.50028505557830799853645049006, 5.44578799312680696564605412015, 6.52062342916023624481267018344, 7.27366030946793406195861630332, 8.80291027341687693723406603485, 9.50968195470761305570455174440, 10.66448331051827243935272483737, 11.23389315536570631838847957523, 11.690456541903226401062843537277, 12.68137854774622284019387151751, 13.77769804730018896386608433455, 14.529235872178475408811056373289, 15.51838651619384101747272444436, 15.63385115107353496103843719627, 16.66739214032938014497744769127, 17.72975682134545292027141266247, 18.90898293148737696071579814437, 19.58501851111087052090355841984, 20.277179127275740193943840817890, 21.07763975594718713996702784550, 21.97193492068366318327759887266