L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s − i·23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.951 − 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s − i·23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (−0.951 − 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2777121720 - 0.5811960859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2777121720 - 0.5811960859i\) |
\(L(1)\) |
\(\approx\) |
\(0.6553686120 - 0.2488228696i\) |
\(L(1)\) |
\(\approx\) |
\(0.6553686120 - 0.2488228696i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.587 - 0.809i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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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
−25.09980785797262828352674841343, −24.33293742937066337100562875990, −22.89912243417044864960093408676, −21.6245037152292643634923834098, −21.46568687078768617063191397788, −20.12071380542753394838236833430, −19.776108161533548198230671380820, −18.89770218027059223960603680008, −17.67345533536038224560582326860, −17.00283527211803551531124684673, −15.99989117734750236785181827584, −15.29391681879035933917860672608, −14.39024589543331984613383545797, −13.0995422465208151246582698540, −12.07441315347451888474363694748, −10.90679715445618855027883119810, −10.26698753151595777950132224553, −9.37693544987063520699434103410, −8.556186193277199833923489454240, −7.72533924944538980603961497813, −6.58244541941763447517247209606, −5.07758351537305404098007043089, −3.8426789188273058237729329544, −2.81630291876660904516742522861, −1.78619788010156418996308645045,
0.448994448014859962234151127613, 2.045709394923979556056969662538, 2.6685420297043904688124997645, 4.4934550546012452346852868891, 6.06584624709271066897012088794, 6.88220387090971772856910972244, 7.64290763375792818484310615217, 8.68400865509975913731655374579, 9.29081108058359972537211638250, 10.453114535570992539259263503352, 11.55489943102639533066644313043, 12.45477545690917414752350566031, 13.574054359747826598717398428890, 14.617469658897360497942853955629, 15.237994837754901568170196808029, 16.411011920163509482055502828423, 17.42378573619467900628526476301, 17.98997134791794904618916031342, 19.04604374544756130264599075912, 19.54487090052960712851827823164, 20.351606173840951495960867502238, 21.28655376003595390171992701011, 22.65332113119264319808382208967, 23.781203996200559888082650453839, 24.44824077672616283603383156852