L(s) = 1 | + (−0.406 − 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (−0.978 + 0.207i)6-s − i·7-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.406 + 0.913i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + i·18-s + (−0.978 − 0.207i)21-s + (−0.207 + 0.978i)22-s + ⋯ |
L(s) = 1 | + (−0.406 − 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (−0.978 + 0.207i)6-s − i·7-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.406 + 0.913i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + i·18-s + (−0.978 − 0.207i)21-s + (−0.207 + 0.978i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1026018048 + 0.01828023723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1026018048 + 0.01828023723i\) |
\(L(1)\) |
\(\approx\) |
\(0.4559936116 - 0.4949628877i\) |
\(L(1)\) |
\(\approx\) |
\(0.4559936116 - 0.4949628877i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.406 - 0.913i)T \) |
| 3 | \( 1 + (0.207 - 0.978i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.406 + 0.913i)T \) |
| 17 | \( 1 + (0.743 - 0.669i)T \) |
| 23 | \( 1 + (-0.994 - 0.104i)T \) |
| 29 | \( 1 + (-0.669 + 0.743i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (-0.104 - 0.994i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.743 - 0.669i)T \) |
| 53 | \( 1 + (0.743 + 0.669i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.207 + 0.978i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.406 + 0.913i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + (0.104 - 0.994i)T \) |
| 97 | \( 1 + (-0.207 + 0.978i)T \) |
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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
−23.51413976271641328246927750852, −22.77816093693307833033357044775, −21.94610137577617203854709121139, −21.15945976883932029102869836623, −20.055192679670888113736585204731, −19.26575400906317013252427367618, −18.16272820785069999904521153417, −17.55951916208614647657835161107, −16.45007278332825744172145690367, −15.77970177038605855773432439088, −15.00267161382753930090181117762, −14.612552784167804859858915891559, −13.30110190917439930434676137317, −12.27481955933586924013438100318, −10.891787298760881090720244848293, −9.95447059261617388443745782968, −9.49527661979827397881238335938, −8.18103402987069810919885388776, −7.902403505515921903979345058057, −6.21489168753279481291744945446, −5.41038796964691229989667692429, −4.73538044715664105183259627087, −3.36488635548971751140035906509, −2.063700504688138190569100876499, −0.0355971083453007943270011208,
0.94877386781738957588642564422, 2.06301242247862486429823911666, 3.07658077766284342881356615102, 4.1203685582564308095130496073, 5.46395879922900811287179216333, 6.952082954662688043137721858234, 7.667391756194623011142768874567, 8.46983352182965359335934705347, 9.59188653192514950554371724372, 10.48372453440403717104338015044, 11.5261723347729562508702796023, 12.1328059950424631449358485159, 13.34284276782804271999703221490, 13.6553942480000027002481219101, 14.59499130004884577137674203006, 16.42773502275560692452888228363, 16.89310349387705430892097158288, 18.01688316525939977800241809030, 18.63670588065451675496141771278, 19.3259035802125751256000119983, 20.24031811099449047331712358137, 20.76141232965637852152431669438, 21.839954159584895873544198891425, 22.85384748107965678462733817945, 23.67244834279610954176480754047