L(s) = 1 | + (7.89 + 4.55i)7-s + (0.383 + 0.221i)11-s + (−9.62 + 5.55i)13-s − 8.01·17-s + 8.11·19-s + (−11.8 − 20.4i)23-s + (−45.9 − 26.5i)29-s + (−14.6 − 25.4i)31-s − 18.4i·37-s + (38.9 − 22.4i)41-s + (−19.9 − 11.5i)43-s + (4.22 − 7.32i)47-s + (17.0 + 29.5i)49-s − 60.5·53-s + (65.9 − 38.0i)59-s + ⋯ |
L(s) = 1 | + (1.12 + 0.651i)7-s + (0.0348 + 0.0201i)11-s + (−0.740 + 0.427i)13-s − 0.471·17-s + 0.427·19-s + (−0.513 − 0.888i)23-s + (−1.58 − 0.913i)29-s + (−0.473 − 0.819i)31-s − 0.499i·37-s + (0.950 − 0.548i)41-s + (−0.463 − 0.267i)43-s + (0.0899 − 0.155i)47-s + (0.348 + 0.602i)49-s − 1.14·53-s + (1.11 − 0.645i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00372 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00372 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.369925540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.369925540\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-7.89 - 4.55i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.383 - 0.221i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.62 - 5.55i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 8.01T + 289T^{2} \) |
| 19 | \( 1 - 8.11T + 361T^{2} \) |
| 23 | \( 1 + (11.8 + 20.4i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (45.9 + 26.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (14.6 + 25.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 18.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-38.9 + 22.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (19.9 + 11.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-4.22 + 7.32i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 60.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-65.9 + 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (2.67 - 4.63i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-95.0 + 54.8i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.35iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-0.792 + 1.37i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (4.22 - 7.32i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-99.7 - 57.6i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−8.404194099774197994678935706326, −7.78390145507489982725167735016, −7.07258742804866585776771976270, −6.05176204062072677799797937439, −5.33849774485079058028754540386, −4.57965624873638936304070978176, −3.78108719468679946462175590011, −2.34534139538323604810331791533, −1.91729106091421642986434671035, −0.31830834830109825940385016031,
1.14022861469020228179360083138, 2.02280971895607046122147940781, 3.24422349821869015758487660968, 4.14304758596364379345397111720, 5.01268297135663945437333995442, 5.54872448762959939034292965246, 6.72427197582153266345368388176, 7.51391706868310235791713729917, 7.88794307660903938693860044499, 8.829064655972105700318218850793