L(s) = 1 | + (−0.678 + 1.17i)2-s + (1.11 + 1.92i)3-s + (0.0800 + 0.138i)4-s + (−1.11 + 1.92i)5-s − 3.02·6-s + (2.60 − 0.437i)7-s − 2.92·8-s + (−0.981 + 1.70i)9-s + (−1.51 − 2.61i)10-s + (0.564 + 0.977i)11-s + (−0.178 + 0.308i)12-s + 2.22·13-s + (−1.25 + 3.36i)14-s − 4.96·15-s + (1.82 − 3.16i)16-s + (−2.02 − 3.51i)17-s + ⋯ |
L(s) = 1 | + (−0.479 + 0.830i)2-s + (0.643 + 1.11i)3-s + (0.0400 + 0.0692i)4-s + (−0.498 + 0.862i)5-s − 1.23·6-s + (0.986 − 0.165i)7-s − 1.03·8-s + (−0.327 + 0.566i)9-s + (−0.477 − 0.827i)10-s + (0.170 + 0.294i)11-s + (−0.0514 + 0.0891i)12-s + 0.617·13-s + (−0.335 + 0.898i)14-s − 1.28·15-s + (0.456 − 0.791i)16-s + (−0.491 − 0.851i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.185742 + 1.25460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.185742 + 1.25460i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.60 + 0.437i)T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (0.678 - 1.17i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.11 - 1.92i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.11 - 1.92i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.564 - 0.977i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + (2.02 + 3.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.80 + 3.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.34 + 2.32i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + (0.172 + 0.299i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.67 - 2.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + (-3.65 + 6.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.17 - 7.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.61 - 4.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.80 - 4.85i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.41 - 7.64i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + (0.407 + 0.706i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.42 + 7.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + (-7.88 + 13.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
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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
−11.83905430778535620895465396541, −11.20657267513683344248015043039, −10.23507117813507083181903811376, −9.091816622472562791868659249792, −8.514020884142501271913062939796, −7.42758155086179532754902343639, −6.75623236107572974200793179763, −5.10080870315216375890931933431, −3.88448479431798720993026874451, −2.83305577190805113889428931608,
1.18760131452415854738669706936, 2.01553587732130678935907064399, 3.65820300316052625142935505530, 5.30391237679442491310631547714, 6.60019304522052790335773064074, 8.027661047314228206857440193148, 8.400336053956682845904963042167, 9.290974021902520640173100524404, 10.67224987593786034254634546742, 11.47479671193510480280017914830