L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.965 − 0.258i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.104 − 0.994i)10-s + (0.629 + 0.777i)11-s + (0.777 + 0.629i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (0.669 + 0.743i)16-s + (−0.777 + 0.629i)17-s + (0.669 − 0.743i)18-s + (−0.0523 + 0.998i)19-s + ⋯ |
L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.965 − 0.258i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.104 − 0.994i)10-s + (0.629 + 0.777i)11-s + (0.777 + 0.629i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (0.669 + 0.743i)16-s + (−0.777 + 0.629i)17-s + (0.669 − 0.743i)18-s + (−0.0523 + 0.998i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.009066770 - 0.5188971767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009066770 - 0.5188971767i\) |
\(L(1)\) |
\(\approx\) |
\(0.9066879643 - 0.4327462116i\) |
\(L(1)\) |
\(\approx\) |
\(0.9066879643 - 0.4327462116i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.207 - 0.978i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.994 - 0.104i)T \) |
| 11 | \( 1 + (0.629 + 0.777i)T \) |
| 13 | \( 1 + (0.891 + 0.453i)T \) |
| 17 | \( 1 + (-0.777 + 0.629i)T \) |
| 19 | \( 1 + (-0.0523 + 0.998i)T \) |
| 23 | \( 1 + (0.978 + 0.207i)T \) |
| 29 | \( 1 + (0.156 - 0.987i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 43 | \( 1 + (-0.951 + 0.309i)T \) |
| 47 | \( 1 + (-0.544 - 0.838i)T \) |
| 53 | \( 1 + (0.933 - 0.358i)T \) |
| 59 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (0.743 - 0.669i)T \) |
| 67 | \( 1 + (-0.358 - 0.933i)T \) |
| 71 | \( 1 + (0.987 - 0.156i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.258 + 0.965i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.998 + 0.0523i)T \) |
| 97 | \( 1 + (0.987 + 0.156i)T \) |
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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
−25.55919891632826692839907192671, −24.65597418435389230951704169089, −23.966980383223581218434038033438, −22.907991902447277772320674606192, −22.15170309074405627282081950429, −21.642574591352757842649936605930, −20.57080816182933946377802878962, −18.77826564749847414228854156713, −18.03382347518712139001982462859, −17.29477920943466504030584747322, −16.57570810656405653635851147840, −15.6995205034052179143427987130, −14.7341890140325207990785845075, −13.488428680742197595363232474, −13.06283104215217701386006344705, −11.5785545446716007270693289600, −10.62974867555587119169095555880, −9.3986035084229133276993347952, −8.688471922336630640818419962420, −6.95878471432597494536864360069, −6.35903687586924542929333718749, −5.47037631565971114962748224010, −4.594149822234073736815295187199, −3.1879636064442960945773139641, −0.98656390314269339893235794925,
1.33776751988161497971924327527, 2.02880921295964704062675366601, 3.84932411267763783205096664233, 4.893320889077348687901433922108, 5.91848976423150247395960813184, 6.75765269348791632169932788436, 8.59365233547729587392867574424, 9.6467961210058126787616218004, 10.47598508681356226093377466365, 11.33445328362322063430999218845, 12.334610417024686735705454176924, 13.08569955759999206597720120385, 13.8992225611571960575905104942, 15.06236333860827718442632663023, 16.58381514023492307624078421153, 17.44736699173548638834728871572, 18.06358893574353098829682249151, 18.943352951683911974507360891861, 20.02101722296824330804839548483, 21.2416090056003711399488954756, 21.54940300323540987979401205760, 22.79581599023540777640416919363, 23.10875964806477414770006590419, 24.37227602624636743190600201462, 25.23016277811430415810629787771