L(s) = 1 | + (−0.5 + 2.59i)7-s + (3.5 − 6.06i)13-s + (−4 + 6.92i)19-s + (2.5 − 4.33i)25-s + 11·31-s + (0.5 − 0.866i)37-s + (6.5 + 11.2i)43-s + (−6.5 − 2.59i)49-s − 61-s + 11·67-s + (5 + 8.66i)73-s − 13·79-s + (14 + 12.1i)91-s + (9.5 + 16.4i)97-s + (3.5 + 6.06i)103-s + ⋯ |
L(s) = 1 | + (−0.188 + 0.981i)7-s + (0.970 − 1.68i)13-s + (−0.917 + 1.58i)19-s + (0.5 − 0.866i)25-s + 1.97·31-s + (0.0821 − 0.142i)37-s + (0.991 + 1.71i)43-s + (−0.928 − 0.371i)49-s − 0.128·61-s + 1.34·67-s + (0.585 + 1.01i)73-s − 1.46·79-s + (1.46 + 1.27i)91-s + (0.964 + 1.67i)97-s + (0.344 + 0.597i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.755963055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.755963055\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 11T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 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
−8.946105476706226060414424613546, −8.145568567701626386864554725086, −7.993711805353816369273444181396, −6.40932566044695748795286546898, −6.08675432134947216703489736872, −5.27393562435221367877755012961, −4.21259013862628706212730304448, −3.19788529700938336785067869900, −2.41704857296097390028232057219, −1.00800530234850543059297618034,
0.77401877661380594419359150400, 2.00369312998558517200722227640, 3.22078743837634814155001143002, 4.24443792208951395271962241392, 4.65437980615657097817887047300, 5.99259733611794745766892944916, 6.84074956637456655038376477234, 7.08458385393307574404798323208, 8.339337352969809099960705614591, 8.925338174027574405539712968071