L(s) = 1 | + (0.447 + 0.894i)2-s + (−0.599 + 0.800i)4-s + (0.0448 − 0.998i)5-s + (0.998 − 0.0598i)7-s + (−0.983 − 0.178i)8-s + (0.913 − 0.406i)10-s + (0.963 + 0.266i)11-s + (−0.691 + 0.722i)13-s + (0.5 + 0.866i)14-s + (−0.280 − 0.959i)16-s + (0.925 + 0.379i)17-s + (0.669 + 0.743i)19-s + (0.772 + 0.635i)20-s + (0.193 + 0.981i)22-s + (−0.309 − 0.951i)23-s + ⋯ |
L(s) = 1 | + (0.447 + 0.894i)2-s + (−0.599 + 0.800i)4-s + (0.0448 − 0.998i)5-s + (0.998 − 0.0598i)7-s + (−0.983 − 0.178i)8-s + (0.913 − 0.406i)10-s + (0.963 + 0.266i)11-s + (−0.691 + 0.722i)13-s + (0.5 + 0.866i)14-s + (−0.280 − 0.959i)16-s + (0.925 + 0.379i)17-s + (0.669 + 0.743i)19-s + (0.772 + 0.635i)20-s + (0.193 + 0.981i)22-s + (−0.309 − 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.574968473 + 1.500818600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.574968473 + 1.500818600i\) |
\(L(1)\) |
\(\approx\) |
\(1.400769173 + 0.5809903172i\) |
\(L(1)\) |
\(\approx\) |
\(1.400769173 + 0.5809903172i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 211 | \( 1 \) |
good | 2 | \( 1 + (0.447 + 0.894i)T \) |
| 5 | \( 1 + (0.0448 - 0.998i)T \) |
| 7 | \( 1 + (0.998 - 0.0598i)T \) |
| 11 | \( 1 + (0.963 + 0.266i)T \) |
| 13 | \( 1 + (-0.691 + 0.722i)T \) |
| 17 | \( 1 + (0.925 + 0.379i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.971 - 0.237i)T \) |
| 31 | \( 1 + (0.0747 - 0.997i)T \) |
| 37 | \( 1 + (0.251 - 0.967i)T \) |
| 41 | \( 1 + (0.999 - 0.0299i)T \) |
| 43 | \( 1 + (0.826 + 0.563i)T \) |
| 47 | \( 1 + (-0.575 + 0.817i)T \) |
| 53 | \( 1 + (0.599 + 0.800i)T \) |
| 59 | \( 1 + (-0.525 - 0.850i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.955 - 0.294i)T \) |
| 79 | \( 1 + (0.983 - 0.178i)T \) |
| 83 | \( 1 + (0.104 + 0.994i)T \) |
| 89 | \( 1 + (-0.936 + 0.351i)T \) |
| 97 | \( 1 + (0.753 - 0.657i)T \) |
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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
−22.40435985744060668144725265503, −21.81132468851076608207282247040, −21.119735503626120871295915011399, −20.057727599522204140870637788064, −19.50062242814601607882225429401, −18.51247315845941812947130790702, −17.89028398396772789219637775688, −17.13491452464396649525074223267, −15.568632648106955855094848088254, −14.73778306533737314464967127084, −14.249098009388461104875571420835, −13.507497061270619857490201551415, −12.20314371572050341925161733607, −11.573571447723372115126949674867, −10.931753334244445827903299111543, −9.98013891203691738343967727178, −9.22222816233982446434934644737, −7.92806006389693336797466963651, −6.98580407880331763138455947154, −5.712246085830158241931375573, −5.02628254637509310020519261850, −3.75363664867416075256352390415, −3.00697335810207503839281343005, −1.93296547037648689480238185924, −0.84128391609853911503848810280,
0.884034448643990490573404103855, 2.10697554294377918736877702266, 3.91784863345444738965495116921, 4.41893060105629248177057297650, 5.3966943290370453068665300642, 6.17878376887013274682431525970, 7.53585958430335723316632177360, 7.97943877670262161477184006685, 9.102515715357984489133373575889, 9.66907994007551788858284086412, 11.356302782245920611995597676992, 12.20366105394834889281889724121, 12.69118724427718161801954213708, 14.10921220428555639039975476116, 14.33158559968776569801583868188, 15.2791448915568231765935868388, 16.59589613718012147064695719174, 16.72559084862008314276576855638, 17.60608851789939989915160719537, 18.52781097440493599774676268020, 19.669069505126041869657368109028, 20.776190250558551899854356206930, 21.16007363342268338915902255570, 22.18868625521124639979564788575, 22.97383449156376811597168066001