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
−20.67363107219580561241959841724, −19.97094614676686888750227772700, −19.24858984610068690417437181019, −18.77916431847334313045346397138, −17.58080572543084329381935751454, −16.94023426403727308861091653511, −16.38809724962498455416557016026, −15.51261991438393713199994888256, −14.536995573451555646813364153472, −14.04530141315416814638897581220, −13.09235010061865028038850121067, −12.22528845980924374316211087139, −11.78561547444969241603683043655, −10.56255340175878528233294396013, −9.85129284299987782949640588241, −9.284171905167699234636335135980, −8.29727755145324906865069423993, −7.15323193420580685298591662305, −6.752675338943174088839203072907, −5.78679060565809488331322244854, −4.667198158490668511036335058547, −3.91658870232361346949989435885, −2.97937104255273545958078728710, −1.962503096095974234508476360075, −0.68295370923381896369422905764,
0.841842221832029447335771538735, 2.178882030847898706323997014541, 3.056153897477754771335884055679, 3.92644945482059534946954481693, 4.91432180742879956361062763567, 5.997894440165738896781359398478, 6.522264913248557610060624345997, 7.463165781577827588522695536382, 8.540945204752140560429224365941, 9.13947374756638030110794101934, 10.07109906088241114222875147131, 10.6539475360477104369507956088, 11.929603547954350694456951986060, 12.31928612022482876422847222932, 13.13549244546311372317027635930, 14.0658810088258994222543830830, 14.837656718658416586600081415501, 15.53736256100140633476917977710, 16.34342095382913893578832131294, 17.20611846618199304764087459650, 17.6217787870234823909776565509, 18.9074601894755200349565034947, 19.4745680933541822030032817313, 19.7298736110823479006171342031, 21.09620009845250531796484215965