L(s) = 1 | + (−0.909 + 0.416i)2-s + (0.653 − 0.757i)4-s + (0.452 + 0.891i)5-s + (0.428 − 0.903i)7-s + (−0.278 + 0.960i)8-s + (−0.783 − 0.621i)10-s + (0.0402 − 0.999i)11-s + (−0.872 − 0.488i)13-s + (−0.0134 + 0.999i)14-s + (−0.147 − 0.989i)16-s + (0.404 − 0.914i)17-s + (0.970 + 0.239i)20-s + (0.379 + 0.925i)22-s + (0.939 + 0.342i)23-s + (−0.589 + 0.807i)25-s + (0.996 + 0.0804i)26-s + ⋯ |
L(s) = 1 | + (−0.909 + 0.416i)2-s + (0.653 − 0.757i)4-s + (0.452 + 0.891i)5-s + (0.428 − 0.903i)7-s + (−0.278 + 0.960i)8-s + (−0.783 − 0.621i)10-s + (0.0402 − 0.999i)11-s + (−0.872 − 0.488i)13-s + (−0.0134 + 0.999i)14-s + (−0.147 − 0.989i)16-s + (0.404 − 0.914i)17-s + (0.970 + 0.239i)20-s + (0.379 + 0.925i)22-s + (0.939 + 0.342i)23-s + (−0.589 + 0.807i)25-s + (0.996 + 0.0804i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2703158913 - 0.8639648367i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2703158913 - 0.8639648367i\) |
\(L(1)\) |
\(\approx\) |
\(0.7525754663 - 0.05964027401i\) |
\(L(1)\) |
\(\approx\) |
\(0.7525754663 - 0.05964027401i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| 53 | \( 1 \) |
good | 2 | \( 1 + (-0.909 + 0.416i)T \) |
| 5 | \( 1 + (0.452 + 0.891i)T \) |
| 7 | \( 1 + (0.428 - 0.903i)T \) |
| 11 | \( 1 + (0.0402 - 0.999i)T \) |
| 13 | \( 1 + (-0.872 - 0.488i)T \) |
| 17 | \( 1 + (0.404 - 0.914i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.303 - 0.952i)T \) |
| 31 | \( 1 + (-0.0402 - 0.999i)T \) |
| 37 | \( 1 + (-0.748 - 0.663i)T \) |
| 41 | \( 1 + (-0.977 - 0.213i)T \) |
| 43 | \( 1 + (0.476 - 0.879i)T \) |
| 47 | \( 1 + (-0.998 + 0.0536i)T \) |
| 59 | \( 1 + (0.956 + 0.291i)T \) |
| 61 | \( 1 + (0.994 + 0.107i)T \) |
| 67 | \( 1 + (-0.982 - 0.186i)T \) |
| 71 | \( 1 + (-0.476 + 0.879i)T \) |
| 73 | \( 1 + (-0.589 - 0.807i)T \) |
| 79 | \( 1 + (0.964 - 0.265i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.897 - 0.440i)T \) |
| 97 | \( 1 + (0.998 + 0.0536i)T \) |
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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
−19.28355967335092273351342211950, −18.37901093099335368722721887818, −17.66983275947325450557611158100, −17.29676035709614623777245825521, −16.54917617668568107130179395225, −15.9055478457607360267361033253, −14.958096368072507648528282299653, −14.5078199126390815258930779458, −13.17837718911050532090534579358, −12.53094287498964602711340215474, −12.16504982901854082368845029825, −11.46409301691557247357898341886, −10.37012957764861017651605552066, −9.89455102087426599703414621720, −9.04471365667619009174380629560, −8.66527955546303597266021245902, −7.89746845818104368988682012632, −6.98805455335668453404575715465, −6.26593508638144729118091048316, −5.062470620363350216244687395695, −4.72182409830534010786132805569, −3.4388946045658738499201428390, −2.44780437270236617656030811798, −1.74366445377468605729203757700, −1.20786651968270603608083148834,
0.22114236502562994682072153840, 0.87317577900769179955653794131, 1.99523093653322004962454546968, 2.799101512656017290506834861898, 3.613202928108163824337595217955, 4.97407679097829173544348656620, 5.59553165306578311187193951146, 6.451816744440477763266511951634, 7.27041977009926999399989598063, 7.568107721436213756848871350885, 8.49364696428999619478063515268, 9.39929277917333388570700823699, 10.087379064584071060219234399075, 10.56594129535567327848587557447, 11.33801728800634304800543976336, 11.81484872939051168407056019910, 13.298548224360280174826574513197, 13.832898529933480994803618025768, 14.53785630836381588089940994830, 15.04877724744570507456085820619, 15.9246218061087782073568789139, 16.71892143854430991884587559737, 17.30389793736248186078917742658, 17.745951858065658012539033127392, 18.590709075734176521418673410552