L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (0.913 + 0.406i)17-s + (0.978 − 0.207i)19-s + (0.809 + 0.587i)20-s + 23-s + (0.669 + 0.743i)25-s + (0.104 + 0.994i)28-s + (0.669 − 0.743i)29-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (0.913 + 0.406i)17-s + (0.978 − 0.207i)19-s + (0.809 + 0.587i)20-s + 23-s + (0.669 + 0.743i)25-s + (0.104 + 0.994i)28-s + (0.669 − 0.743i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2933012989 - 1.098945458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2933012989 - 1.098945458i\) |
\(L(1)\) |
\(\approx\) |
\(0.6696000952 - 0.6054552896i\) |
\(L(1)\) |
\(\approx\) |
\(0.6696000952 - 0.6054552896i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.913 - 0.406i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.978 - 0.207i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (0.978 + 0.207i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.669 - 0.743i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.913 - 0.406i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.104 + 0.994i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−21.5994012171073113115520545756, −20.64304639383739073368528332724, −19.46325925850231543057324108608, −18.90612089766045358699682912096, −18.28453842759062452702815772805, −17.55261686935102659206609363228, −16.30998072233471642252018516819, −16.05503545986959017201634694644, −15.26999240482230227829427222214, −14.54976206292839662184547702428, −13.961885663988343850360962383500, −12.70379477063626462235300803675, −12.26062829978575385706691267922, −11.37615851741110647371727048124, −10.2431750141248653454282729629, −9.29816300091867112777157524106, −8.63418450878328266274858037831, −7.72242573215335919908061718628, −7.13391969324835936918625857435, −6.22796802736198711208664542817, −5.349217903300791081153100368123, −4.61711303596843686883156836703, −3.377438506546772439469364077315, −2.92658581215919564111183848083, −1.00775789455930083140538941316,
0.61620437224985911444796194024, 1.31834269087394118179962772920, 2.863841137339344086873298085460, 3.533045395911440812782290939622, 4.34652744017131152776951489782, 5.03227716317928908229749235703, 6.21088551513342358795252028433, 7.52333440108238680072485679248, 8.00144309146885721301413780355, 9.10102091564046848837543165674, 9.84334438291669689313290294985, 10.617501914708548469751577604870, 11.49103781007006841074449293260, 11.98015639005782313170198939789, 13.01965162951435474624848745569, 13.39336043412834505763885267945, 14.45500181314188230259762617002, 15.154958070832862438898325466874, 16.2950334690528906933243997691, 16.866939631975346497083085661557, 17.739546062649006632529266295647, 18.71812004592229139907618121245, 19.41179732322101488498269702405, 19.84861883819688362399485518089, 20.714464359830029912920816812635