L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 − 2.59i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 3·6-s − 0.999·8-s + (−3 − 5.19i)9-s + 0.999·10-s + 2·11-s + (1.50 + 2.59i)12-s + (0.5 − 0.866i)13-s + (−1.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (3 − 5.19i)18-s + (−1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 − 1.49i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 1.22·6-s − 0.353·8-s + (−1 − 1.73i)9-s + 0.316·10-s + 0.603·11-s + (0.433 + 0.749i)12-s + (0.138 − 0.240i)13-s + (−0.387 − 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (0.707 − 1.22i)18-s + (−0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.93869 - 0.766476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93869 - 0.766476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (5.5 - 2.59i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95163200795000568405637685859, −10.19166817942503132232718048092, −8.946564042132285791414141588271, −8.299301297591984611693436244611, −7.57527665202680956664320514258, −6.52228201100071469565806930147, −5.88171599656515378543321409274, −4.21201875262721930949205950758, −2.84862013964375338544521413445, −1.40283922902562266868125513243,
2.30393138872681614184910660274, 3.42054882189283528330028759162, 4.23807683064305712673795393728, 5.23746815707710114612601892896, 6.58063465243093294159014663586, 8.188868981264419101140333349812, 9.045073460082600879588754395068, 9.857731872583700502391303430650, 10.38904104770267075409135561590, 11.32321448255447855739248646614