L(s) = 1 | + (−0.743 + 0.669i)2-s + (0.965 + 0.258i)3-s + (0.104 − 0.994i)4-s + (−0.406 − 0.913i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.913 + 0.406i)10-s + (0.933 − 0.358i)11-s + (0.358 − 0.933i)12-s + (0.453 + 0.891i)13-s + (−0.156 − 0.987i)15-s + (−0.978 − 0.207i)16-s + (−0.358 − 0.933i)17-s + (−0.978 + 0.207i)18-s + (0.544 + 0.838i)19-s + ⋯ |
L(s) = 1 | + (−0.743 + 0.669i)2-s + (0.965 + 0.258i)3-s + (0.104 − 0.994i)4-s + (−0.406 − 0.913i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.913 + 0.406i)10-s + (0.933 − 0.358i)11-s + (0.358 − 0.933i)12-s + (0.453 + 0.891i)13-s + (−0.156 − 0.987i)15-s + (−0.978 − 0.207i)16-s + (−0.358 − 0.933i)17-s + (−0.978 + 0.207i)18-s + (0.544 + 0.838i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.936797423 + 0.08145994171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.936797423 + 0.08145994171i\) |
\(L(1)\) |
\(\approx\) |
\(1.109723534 + 0.1593189164i\) |
\(L(1)\) |
\(\approx\) |
\(1.109723534 + 0.1593189164i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.743 + 0.669i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (-0.406 - 0.913i)T \) |
| 11 | \( 1 + (0.933 - 0.358i)T \) |
| 13 | \( 1 + (0.453 + 0.891i)T \) |
| 17 | \( 1 + (-0.358 - 0.933i)T \) |
| 19 | \( 1 + (0.544 + 0.838i)T \) |
| 23 | \( 1 + (-0.669 - 0.743i)T \) |
| 29 | \( 1 + (0.987 - 0.156i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (0.951 + 0.309i)T \) |
| 47 | \( 1 + (0.0523 - 0.998i)T \) |
| 53 | \( 1 + (0.629 + 0.777i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.207 - 0.978i)T \) |
| 67 | \( 1 + (0.777 - 0.629i)T \) |
| 71 | \( 1 + (0.156 - 0.987i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.258 + 0.965i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.838 - 0.544i)T \) |
| 97 | \( 1 + (-0.156 - 0.987i)T \) |
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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
−25.7414600043214065348422731692, −24.72809551700157781662576341235, −23.49640317651867477200411096204, −22.221411265952683495388313413, −21.65430006909706422572770766172, −20.324259154445908317848955705390, −19.760241861434920856512993972824, −19.17664946560020435785605373767, −18.04014023346639351990902359281, −17.61785889732676265791144981233, −15.9642299839213501803634377013, −15.18524557617085038086700653685, −14.13739452077933910755578995850, −13.103004919574699047757667518652, −12.12338881590480449286784074000, −11.08492769088482726162128084175, −10.13809242872079181773737888989, −9.18528389383939817557193475970, −8.19591010751536946705474394301, −7.3842328846689140566086346912, −6.45890739201787453989996441184, −4.10042515739882440366392659162, −3.34632473160947415930189091013, −2.34253803625794034620800560754, −1.08231336871330124850413892227,
0.84455368318739879645431923616, 2.02451324688887499424542801963, 3.846523651153645692780928852123, 4.7607773083092600890947628502, 6.19660449224363319810538430671, 7.406082700073462992904429146706, 8.30801112735572518977272178917, 9.09379879981961670897696005896, 9.64286750382885375870564826724, 11.070521086304105275617416278353, 12.19897672730591591735301235870, 13.73149586443555751109199115861, 14.23018128744845115279129548363, 15.37516732946755651610887225353, 16.34345123670999410095029598406, 16.59516984696848656311160509836, 18.15523935703351312728929469830, 18.96494163114303609137298798475, 19.92250197249453715794963016329, 20.36105206495731296963735406034, 21.48421345286467044953782943716, 22.82753456708646285988576938312, 23.965899196353615055259456758612, 24.68799471494244506848717206784, 25.17651013615362460822791034858