L(s) = 1 | + (1.19 − 0.757i)2-s + (−0.755 + 0.654i)3-s + (0.853 − 1.80i)4-s + (−2.01 + 0.290i)5-s + (−0.406 + 1.35i)6-s + (0.214 − 0.470i)7-s + (−0.349 − 2.80i)8-s + (0.142 − 0.989i)9-s + (−2.19 + 1.87i)10-s + (0.754 − 2.56i)11-s + (0.539 + 1.92i)12-s + (5.11 − 2.33i)13-s + (−0.0994 − 0.723i)14-s + (1.33 − 1.54i)15-s + (−2.54 − 3.08i)16-s + (−3.23 − 2.07i)17-s + ⋯ |
L(s) = 1 | + (0.844 − 0.535i)2-s + (−0.436 + 0.378i)3-s + (0.426 − 0.904i)4-s + (−0.902 + 0.129i)5-s + (−0.166 + 0.552i)6-s + (0.0811 − 0.177i)7-s + (−0.123 − 0.992i)8-s + (0.0474 − 0.329i)9-s + (−0.693 + 0.593i)10-s + (0.227 − 0.774i)11-s + (0.155 + 0.555i)12-s + (1.41 − 0.647i)13-s + (−0.0265 − 0.193i)14-s + (0.344 − 0.398i)15-s + (−0.635 − 0.771i)16-s + (−0.784 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.999393 - 1.28307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999393 - 1.28307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 + 0.757i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (2.90 + 3.81i)T \) |
good | 5 | \( 1 + (2.01 - 0.290i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.214 + 0.470i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.754 + 2.56i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.11 + 2.33i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (3.23 + 2.07i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.853 + 1.32i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.80 + 2.80i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.62 + 3.02i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-6.67 - 0.960i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.979 - 6.81i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.51 - 6.50i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 + (-1.29 - 0.592i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.925 + 0.422i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.46 - 3.87i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.456 - 1.55i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (14.2 - 4.17i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 6.62i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.77 + 8.27i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (5.36 + 0.770i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-8.92 - 10.3i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.0860 + 0.598i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.91114563584055396742339554942, −10.02548742384133772720205653216, −8.832446298048733888201629613238, −7.83327986956967015851063504713, −6.46876008810809761707172333839, −5.90880288238686708016801963680, −4.54463786122056574933305029196, −3.92336472003812302677864864053, −2.85031936015593467139377463617, −0.78138007725372602549412245749,
1.93663957147117728274204132181, 3.74035803055515161817166834858, 4.33443195866118155777520615738, 5.57393401506599525977973892981, 6.49053478912925792395246694796, 7.22636851643099738917275428619, 8.201319361314873228595145350280, 8.874843947110382615038972499060, 10.48959857961111671587091441514, 11.46843227320427311849640942487