L(s) = 1 | + (1.36 + 2.36i)5-s + (−0.931 + 2.47i)7-s + (−1.17 + 2.04i)11-s + 6.68·13-s + (2.09 − 3.63i)17-s + (−1.80 − 3.11i)19-s + (3.40 + 5.90i)23-s + (−1.23 + 2.13i)25-s − 1.00·29-s + (−1.05 + 1.82i)31-s + (−7.13 + 1.17i)35-s + (2.47 + 4.29i)37-s − 8.91·41-s − 2.25·43-s + (5.85 + 10.1i)47-s + ⋯ |
L(s) = 1 | + (0.610 + 1.05i)5-s + (−0.352 + 0.935i)7-s + (−0.355 + 0.615i)11-s + 1.85·13-s + (0.508 − 0.881i)17-s + (−0.413 − 0.715i)19-s + (0.710 + 1.23i)23-s + (−0.246 + 0.426i)25-s − 0.186·29-s + (−0.189 + 0.327i)31-s + (−1.20 + 0.199i)35-s + (0.407 + 0.706i)37-s − 1.39·41-s − 0.343·43-s + (0.853 + 1.47i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.823634822\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.823634822\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.931 - 2.47i)T \) |
good | 5 | \( 1 + (-1.36 - 2.36i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.17 - 2.04i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 + (-2.09 + 3.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.80 + 3.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.40 - 5.90i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 + (1.05 - 1.82i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.47 - 4.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.91T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 + (-5.85 - 10.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.13 - 7.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.91 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.00 + 10.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.03 + 5.25i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 + (-2.28 + 3.96i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.252 - 0.437i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + (-4.22 - 7.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−9.520195882162496892058663335253, −9.132260860494195641033296896888, −8.092732873931718515467036708237, −7.12268924800866344610582736390, −6.37626717622924025223013529236, −5.77213936695705964563659030626, −4.81615644846184978034655805030, −3.34197672292601282561534931894, −2.80316610804178853322130161265, −1.58385892550694854229078788275,
0.76507829935207486510080387595, 1.71928211349230438129959879187, 3.40668487466838139521213483374, 4.05865006999489004015271121179, 5.19203988481934877810782024758, 6.00852873620388987031070609012, 6.61022608723782466511133329170, 7.900958357438526743789831775284, 8.546742573271565045332257533143, 9.069116934390451538202198439308