L(s) = 1 | + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 4·13-s + 1.99·15-s + (3 + 5.19i)17-s + (4 − 6.92i)19-s + (3 − 5.19i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s − 10·29-s + (2 + 3.46i)31-s + (0.999 − 1.73i)33-s + (−3 + 5.19i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 1.10·13-s + 0.516·15-s + (0.727 + 1.26i)17-s + (0.917 − 1.58i)19-s + (0.625 − 1.08i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s − 1.85·29-s + (0.359 + 0.622i)31-s + (0.174 − 0.301i)33-s + (−0.493 + 0.854i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80707 - 0.114677i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80707 - 0.114677i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.72778583873254232817321505124, −9.688946276586426160804765139668, −8.835877260878964498105470781262, −8.412426757454182326040029176675, −7.14265581430412680312797508121, −5.85860392646274529244502887464, −5.18891765822095017483420416119, −4.01716907375447724638111147930, −2.91728336039430706781360995339, −1.23553727931513015429673319306,
1.48454647706044124862202797041, 2.82157880809136794235737109534, 3.78106439838390656310252233410, 5.47156165595483659599096975733, 6.11892232742314159672942746422, 7.45629129092727535273265018336, 7.63984826694279960957230548952, 9.142962783455452026301481432854, 9.731319955720249993534157848566, 10.73965459330473160920261018699