L(s) = 1 | + (674. + 389. i)5-s + (−1.63e3 + 1.75e3i)7-s + (−291. + 168. i)11-s + 46.7·13-s + (−9.81e4 + 5.66e4i)17-s + (4.11e4 − 7.13e4i)19-s + (−3.26e5 − 1.88e5i)23-s + (1.07e5 + 1.87e5i)25-s + 7.47e5i·29-s + (−7.74e5 − 1.34e6i)31-s + (−1.78e6 + 5.49e5i)35-s + (4.75e5 − 8.23e5i)37-s − 3.98e6i·41-s − 1.78e5·43-s + (4.60e6 + 2.66e6i)47-s + ⋯ |
L(s) = 1 | + (1.07 + 0.623i)5-s + (−0.680 + 0.732i)7-s + (−0.0199 + 0.0115i)11-s + 0.00163·13-s + (−1.17 + 0.678i)17-s + (0.316 − 0.547i)19-s + (−1.16 − 0.673i)23-s + (0.276 + 0.478i)25-s + 1.05i·29-s + (−0.838 − 1.45i)31-s + (−1.19 + 0.366i)35-s + (0.253 − 0.439i)37-s − 1.41i·41-s − 0.0523·43-s + (0.944 + 0.545i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6970671515\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6970671515\) |
\(L(5)\) |
|
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 | \( 1 + (1.63e3 - 1.75e3i)T \) |
good | 5 | \( 1 + (-674. - 389. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (291. - 168. i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 46.7T + 8.15e8T^{2} \) |
| 17 | \( 1 + (9.81e4 - 5.66e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-4.11e4 + 7.13e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.26e5 + 1.88e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 7.47e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (7.74e5 + 1.34e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-4.75e5 + 8.23e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.98e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.78e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.60e6 - 2.66e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-7.08e5 + 4.09e5i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-8.19e6 + 4.72e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.08e7 - 1.87e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.78e6 - 1.34e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.95e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.59e7 - 2.75e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.44e7 + 4.23e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 9.26e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.56e7 + 2.05e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 6.33e7T + 7.83e15T^{2} \) |
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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
−10.28800222860595830771575998967, −9.404120144340305452352712435558, −8.649600485975053337018297867221, −7.14649757329820727311837133183, −6.22101629995065747692960031900, −5.58750359086527514149624225475, −4.02703776719189731176078480288, −2.61927419156904250366870582281, −1.98201855502651119226802709561, −0.14431223891684396650387398873,
1.12612012441197187164270157833, 2.23676311054981734249092429424, 3.64793834528962798537735880179, 4.84987334049250444198169493257, 5.92441557207941275766861795844, 6.80607417792898111453863129285, 7.999376371265666205012686763370, 9.271644463122314746355598141144, 9.756508945444854082250389485108, 10.71351375209328319858660918213