L(s) = 1 | + 1.98i·2-s − 1.94·4-s + 3.55·5-s + 0.0996i·8-s + 7.06i·10-s − 4.53i·11-s + 4.71i·13-s − 4.09·16-s + 7.51·17-s + 1.30i·19-s − 6.92·20-s + 9.01·22-s − 0.567i·23-s + 7.62·25-s − 9.37·26-s + ⋯ |
L(s) = 1 | + 1.40i·2-s − 0.974·4-s + 1.58·5-s + 0.0352i·8-s + 2.23i·10-s − 1.36i·11-s + 1.30i·13-s − 1.02·16-s + 1.82·17-s + 0.298i·19-s − 1.54·20-s + 1.92·22-s − 0.118i·23-s + 1.52·25-s − 1.83·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.709783626\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.709783626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.98iT - 2T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 11 | \( 1 + 4.53iT - 11T^{2} \) |
| 13 | \( 1 - 4.71iT - 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 - 1.30iT - 19T^{2} \) |
| 23 | \( 1 + 0.567iT - 23T^{2} \) |
| 29 | \( 1 - 8.53iT - 29T^{2} \) |
| 31 | \( 1 + 7.69iT - 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.85iT - 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 0.664iT - 61T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 + 4.70iT - 71T^{2} \) |
| 73 | \( 1 + 5.57iT - 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−9.047516008568312296273572344181, −8.089654833055735620173228082103, −7.37324457565035926378924241399, −6.50835470829235217518218038940, −5.91047413049866980616042937343, −5.61394448536978760197830509189, −4.73288396298210367288725343862, −3.47998281919504839484032511907, −2.35263902298456815302700473891, −1.24290649892300836488399180252,
0.951298387268274562729135677944, 1.79461902039964229831997805675, 2.59829678461889895769917433964, 3.27634348683252114085381860629, 4.44111792095183889696722601870, 5.32538291292897329343203592246, 5.91419160645120450437234284659, 6.94404987146913206258475191386, 7.74257679152861048964039631481, 8.797099291383321438729981418961