L(s) = 1 | + (−2 − 1.73i)7-s + (−1 + 1.73i)13-s + (0.5 + 0.866i)19-s − 5·25-s + (3.5 + 6.06i)31-s + (5 + 8.66i)37-s + (−2.5 − 4.33i)43-s + (1.00 + 6.92i)49-s + (0.5 − 0.866i)61-s + (8 + 13.8i)67-s + (−8.5 + 14.7i)73-s + (2 − 3.46i)79-s + (5 − 1.73i)91-s + (9.5 + 16.4i)97-s + 20·103-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)7-s + (−0.277 + 0.480i)13-s + (0.114 + 0.198i)19-s − 25-s + (0.628 + 1.08i)31-s + (0.821 + 1.42i)37-s + (−0.381 − 0.660i)43-s + (0.142 + 0.989i)49-s + (0.0640 − 0.110i)61-s + (0.977 + 1.69i)67-s + (−0.994 + 1.72i)73-s + (0.225 − 0.389i)79-s + (0.524 − 0.181i)91-s + (0.964 + 1.67i)97-s + 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.007679373\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007679373\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.291122328611718349895599706417, −8.425267446671817411179928971512, −7.61218400205677995047146541298, −6.82927548833839094404464875102, −6.25246162473715572955443888923, −5.22556683155357210348381521104, −4.28581574043032567823460635352, −3.51342737769171320840106795590, −2.51000669199540901406570886824, −1.16353929674270346245792418559,
0.37259392282041021892127344074, 2.06653202313522153868653093689, 2.93978152154037080513733695321, 3.87201055401858139275699630868, 4.89847339124623733294036718634, 5.85281977791787056277457947149, 6.30817956212715710106039443967, 7.40088258895552779700720193226, 8.007623087298332746234245775671, 8.958011323934146674340728859508