L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (−0.5 + 0.866i)11-s + 2·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + (3 + 5.19i)19-s + 0.999·20-s − 0.999·22-s + (3.5 + 6.06i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (0.158 − 0.273i)10-s + (−0.150 + 0.261i)11-s + 0.554·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + (0.688 + 1.19i)19-s + 0.223·20-s − 0.213·22-s + (0.729 + 1.26i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.665327604\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665327604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−9.722942406712210365253753669600, −8.803437196083605841855088858414, −8.026861769763379612240257243805, −7.16336257771782792552786202175, −6.55421423571550900349679512308, −5.50847474699024380613983410841, −4.84087452066317297379660552526, −3.64370442573907985162946833372, −3.03836291788202732639667413891, −1.07291782668336743015093228662,
0.77161620851336411024148920747, 2.46910547395423037472204259919, 3.21729307601242294853268920414, 4.05529041732218200274643882425, 5.21766004599126879123267303568, 6.10146751378147095158105901835, 6.78158456778465121390164540720, 7.83387624338365706137608638082, 8.912745131488240073196171624940, 9.384282573028651789534262960905