L(s) = 1 | + 0.343·5-s + (2.14 − 1.55i)7-s + 4.91·11-s + (0.974 + 1.68i)13-s + (2.07 + 3.59i)17-s + (0.202 − 0.350i)19-s + 2.75·23-s − 4.88·25-s + (1.63 − 2.83i)29-s + (1.64 − 2.84i)31-s + (0.736 − 0.533i)35-s + (−3.38 + 5.87i)37-s + (−2.81 − 4.87i)41-s + (−4.96 + 8.59i)43-s + (6.60 + 11.4i)47-s + ⋯ |
L(s) = 1 | + 0.153·5-s + (0.810 − 0.586i)7-s + 1.48·11-s + (0.270 + 0.467i)13-s + (0.502 + 0.871i)17-s + (0.0464 − 0.0803i)19-s + 0.575·23-s − 0.976·25-s + (0.304 − 0.527i)29-s + (0.295 − 0.511i)31-s + (0.124 − 0.0901i)35-s + (−0.557 + 0.965i)37-s + (−0.439 − 0.760i)41-s + (−0.756 + 1.31i)43-s + (0.962 + 1.66i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.331368808\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.331368808\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.14 + 1.55i)T \) |
good | 5 | \( 1 - 0.343T + 5T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 + (-0.974 - 1.68i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.07 - 3.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.202 + 0.350i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 + (-1.63 + 2.83i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.64 + 2.84i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.38 - 5.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 + 4.87i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.96 - 8.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.60 - 11.4i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.38 - 2.39i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.22 + 7.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.37 + 9.31i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.21 + 7.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.08T + 71T^{2} \) |
| 73 | \( 1 + (-4.58 - 7.93i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.24 + 3.89i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.62 + 8.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.54 + 6.13i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.31 - 2.27i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.027709810575488066024544691275, −8.215352115229133053390656403416, −7.56911235793509865802239602610, −6.57089861126565383475015585665, −6.08391257684383702190199822017, −4.89585171186591774614662688618, −4.15715992831239278362745897297, −3.43168274693255524804599065871, −1.90109390078348733104043078930, −1.12945739400494823974472594182,
1.05729006709867879411382822688, 2.06907201367401830813800157370, 3.26213685880434849726955723024, 4.14723030545703897747140726272, 5.20589176025265334383222400141, 5.72193397039326467153738468035, 6.78855335936597472755560721983, 7.39687977570800388800098954224, 8.505731136006064067772316524714, 8.850101280158688940309017789866