L(s) = 1 | + (0.904 − 0.426i)2-s + (0.716 + 0.697i)3-s + (0.635 − 0.771i)4-s + (0.451 − 0.892i)5-s + (0.945 + 0.324i)6-s + (−0.677 + 0.735i)7-s + (0.245 − 0.969i)8-s + (0.0275 + 0.999i)9-s + (0.0275 − 0.999i)10-s + (0.851 − 0.523i)11-s + (0.993 − 0.110i)12-s + (0.137 + 0.990i)13-s + (−0.298 + 0.954i)14-s + (0.945 − 0.324i)15-s + (−0.191 − 0.981i)16-s + (−0.191 − 0.981i)17-s + ⋯ |
L(s) = 1 | + (0.904 − 0.426i)2-s + (0.716 + 0.697i)3-s + (0.635 − 0.771i)4-s + (0.451 − 0.892i)5-s + (0.945 + 0.324i)6-s + (−0.677 + 0.735i)7-s + (0.245 − 0.969i)8-s + (0.0275 + 0.999i)9-s + (0.0275 − 0.999i)10-s + (0.851 − 0.523i)11-s + (0.993 − 0.110i)12-s + (0.137 + 0.990i)13-s + (−0.298 + 0.954i)14-s + (0.945 − 0.324i)15-s + (−0.191 − 0.981i)16-s + (−0.191 − 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.175605446 - 0.6375253173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.175605446 - 0.6375253173i\) |
\(L(1)\) |
\(\approx\) |
\(2.256936745 - 0.3304294247i\) |
\(L(1)\) |
\(\approx\) |
\(2.256936745 - 0.3304294247i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.904 - 0.426i)T \) |
| 3 | \( 1 + (0.716 + 0.697i)T \) |
| 5 | \( 1 + (0.451 - 0.892i)T \) |
| 7 | \( 1 + (-0.677 + 0.735i)T \) |
| 11 | \( 1 + (0.851 - 0.523i)T \) |
| 13 | \( 1 + (0.137 + 0.990i)T \) |
| 17 | \( 1 + (-0.191 - 0.981i)T \) |
| 19 | \( 1 + (0.716 + 0.697i)T \) |
| 23 | \( 1 + (0.245 + 0.969i)T \) |
| 29 | \( 1 + (-0.926 - 0.376i)T \) |
| 31 | \( 1 + (0.945 + 0.324i)T \) |
| 37 | \( 1 + (0.137 - 0.990i)T \) |
| 41 | \( 1 + (0.451 - 0.892i)T \) |
| 43 | \( 1 + (0.0275 - 0.999i)T \) |
| 47 | \( 1 + (-0.754 + 0.656i)T \) |
| 53 | \( 1 + (0.904 + 0.426i)T \) |
| 59 | \( 1 + (-0.677 + 0.735i)T \) |
| 61 | \( 1 + (-0.821 - 0.569i)T \) |
| 67 | \( 1 + (-0.821 + 0.569i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.926 + 0.376i)T \) |
| 79 | \( 1 + (0.350 - 0.936i)T \) |
| 83 | \( 1 + (-0.754 + 0.656i)T \) |
| 89 | \( 1 + (-0.191 - 0.981i)T \) |
| 97 | \( 1 + (0.635 - 0.771i)T \) |
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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
−23.09642825691534518661665755552, −22.68232954112713332034670406878, −21.900450982226554899369549537353, −20.75855830746891124905302554464, −19.998633502208509526404768550623, −19.415513332595940371051428608227, −18.14254301875861068377859019332, −17.4600734817012531538576303194, −16.57866246195149373462425360673, −15.12283654540329342490742716225, −14.92523668870236917798961889363, −13.89089664425554853282507072577, −13.261997051179372574234741669210, −12.65802130182660975140442465824, −11.52606754180081804927860594351, −10.44854599951606434530770860378, −9.44467509461805359541836532991, −8.160077207710713732249801802770, −7.2824343510248309477633007245, −6.59173431191958399802380793668, −6.0342816202662772546966145256, −4.40971587580685367352063494857, −3.34328595473111887808099101408, −2.81109942775951548156826280353, −1.51191248254768197004314623588,
1.4476088303246715643724621200, 2.46821862469289574960306931159, 3.5108621485954745067330469591, 4.30448758401837819491224610090, 5.36231117404957401261728526744, 6.03649723846260419868077789430, 7.36290780164907076797049753184, 9.04009001687244683283978962350, 9.21367654156375673696474043746, 10.13003993207027285351008344694, 11.49449378421552186397948019655, 12.06210032456478532910978791861, 13.23257935278280195035108951724, 13.82855255725960344609121720082, 14.48614131153843096768139106936, 15.72834279764424848771577076308, 16.08576180249081746474198710585, 16.9435034261042199806176025135, 18.610619914639243329244271838561, 19.37084222311077759060676182422, 20.025405468444555718390290107170, 20.97691048069730349881687830561, 21.38377112297640248191366143488, 22.20121783451732401536289159492, 22.861721107416636772268540807764