L(s) = 1 | + (0.336 − 0.583i)2-s + (1.77 + 3.07i)4-s + (−1.93 − 1.11i)5-s + (−6.82 + 1.55i)7-s + 5.08·8-s + (−1.30 + 0.752i)10-s + (−0.0223 − 0.0387i)11-s + 23.0i·13-s + (−1.38 + 4.50i)14-s + (−5.38 + 9.32i)16-s + (8.16 − 4.71i)17-s + (0.991 + 0.572i)19-s − 7.93i·20-s − 0.0301·22-s + (−22.1 + 38.3i)23-s + ⋯ |
L(s) = 1 | + (0.168 − 0.291i)2-s + (0.443 + 0.767i)4-s + (−0.387 − 0.223i)5-s + (−0.974 + 0.222i)7-s + 0.635·8-s + (−0.130 + 0.0752i)10-s + (−0.00203 − 0.00352i)11-s + 1.76i·13-s + (−0.0992 + 0.321i)14-s + (−0.336 + 0.582i)16-s + (0.480 − 0.277i)17-s + (0.0521 + 0.0301i)19-s − 0.396i·20-s − 0.00137·22-s + (−0.961 + 1.66i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0974 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0974 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.877306 + 0.967415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877306 + 0.967415i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (6.82 - 1.55i)T \) |
good | 2 | \( 1 + (-0.336 + 0.583i)T + (-2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (0.0223 + 0.0387i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 23.0iT - 169T^{2} \) |
| 17 | \( 1 + (-8.16 + 4.71i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.991 - 0.572i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (22.1 - 38.3i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 53.0T + 841T^{2} \) |
| 31 | \( 1 + (-19.5 + 11.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (21.1 - 36.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 38.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 76.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-23.5 - 13.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-9.49 - 16.4i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-4.21 + 2.43i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.6 + 19.4i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-3.50 - 6.06i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 46.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-72.3 + 41.7i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.2 - 17.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 125. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (40.4 + 23.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 3.11iT - 9.40e3T^{2} \) |
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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
−11.85404050621241572937512779429, −11.06612176956912307487765712614, −9.694878722367706250039357860874, −9.009730883964101703063285313103, −7.70704586227643186353207600732, −7.00274278018513507596927194056, −5.83823385043584015538769722174, −4.20379691378134948666829796084, −3.42523150853770625078864138850, −1.97773098533564910692859584198,
0.56779936941290883257075542879, 2.61930884512760112830373891300, 3.92387936273947543136933607259, 5.45748647870396266554201944208, 6.19693495105786354606341761860, 7.23329827529742786859611212896, 8.105480201104380444344183555289, 9.547652261439355222696484624276, 10.42356832264941685474646159707, 10.86780829673897436084522747514