L(s) = 1 | + (−0.909 − 0.415i)7-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.989 − 0.142i)37-s + (0.142 + 0.989i)41-s + (0.755 − 0.654i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.909 + 0.415i)53-s + (−0.415 − 0.909i)59-s + (0.654 − 0.755i)61-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)7-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.989 − 0.142i)37-s + (0.142 + 0.989i)41-s + (0.755 − 0.654i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.909 + 0.415i)53-s + (−0.415 − 0.909i)59-s + (0.654 − 0.755i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.267508683 - 0.2550034939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.267508683 - 0.2550034939i\) |
\(L(1)\) |
\(\approx\) |
\(0.9865190473 - 0.05682850012i\) |
\(L(1)\) |
\(\approx\) |
\(0.9865190473 - 0.05682850012i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (-0.909 - 0.415i)T \) |
| 11 | \( 1 + (0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.909 + 0.415i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (-0.989 - 0.142i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.755 - 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.909 + 0.415i)T \) |
| 59 | \( 1 + (-0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.281 + 0.959i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (0.540 + 0.841i)T \) |
| 79 | \( 1 + (-0.415 - 0.909i)T \) |
| 83 | \( 1 + (0.989 + 0.142i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.989 - 0.142i)T \) |
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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
−21.09620009845250531796484215965, −19.7298736110823479006171342031, −19.4745680933541822030032817313, −18.9074601894755200349565034947, −17.6217787870234823909776565509, −17.20611846618199304764087459650, −16.34342095382913893578832131294, −15.53736256100140633476917977710, −14.837656718658416586600081415501, −14.0658810088258994222543830830, −13.13549244546311372317027635930, −12.31928612022482876422847222932, −11.929603547954350694456951986060, −10.6539475360477104369507956088, −10.07109906088241114222875147131, −9.13947374756638030110794101934, −8.540945204752140560429224365941, −7.463165781577827588522695536382, −6.522264913248557610060624345997, −5.997894440165738896781359398478, −4.91432180742879956361062763567, −3.92644945482059534946954481693, −3.056153897477754771335884055679, −2.178882030847898706323997014541, −0.841842221832029447335771538735,
0.68295370923381896369422905764, 1.962503096095974234508476360075, 2.97937104255273545958078728710, 3.91658870232361346949989435885, 4.667198158490668511036335058547, 5.78679060565809488331322244854, 6.752675338943174088839203072907, 7.15323193420580685298591662305, 8.29727755145324906865069423993, 9.284171905167699234636335135980, 9.85129284299987782949640588241, 10.56255340175878528233294396013, 11.78561547444969241603683043655, 12.22528845980924374316211087139, 13.09235010061865028038850121067, 14.04530141315416814638897581220, 14.536995573451555646813364153472, 15.51261991438393713199994888256, 16.38809724962498455416557016026, 16.94023426403727308861091653511, 17.58080572543084329381935751454, 18.77916431847334313045346397138, 19.24858984610068690417437181019, 19.97094614676686888750227772700, 20.67363107219580561241959841724