| L(s) = 1 | + (−8.95 + 8.95i)3-s + (−127. + 127. i)5-s + 458.·7-s + 568. i·9-s + (−912. − 912. i)11-s + (−1.92e3 − 1.92e3i)13-s − 2.27e3i·15-s − 1.61e3·17-s + (784. − 784. i)19-s + (−4.10e3 + 4.10e3i)21-s − 1.12e4·23-s − 1.67e4i·25-s + (−1.16e4 − 1.16e4i)27-s + (−2.32e4 − 2.32e4i)29-s − 2.45e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.331 + 0.331i)3-s + (−1.01 + 1.01i)5-s + 1.33·7-s + 0.780i·9-s + (−0.685 − 0.685i)11-s + (−0.876 − 0.876i)13-s − 0.674i·15-s − 0.329·17-s + (0.114 − 0.114i)19-s + (−0.443 + 0.443i)21-s − 0.926·23-s − 1.07i·25-s + (−0.590 − 0.590i)27-s + (−0.952 − 0.952i)29-s − 0.823i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0213803 - 0.0611427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0213803 - 0.0611427i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (8.95 - 8.95i)T - 729iT^{2} \) |
| 5 | \( 1 + (127. - 127. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 - 458.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (912. + 912. i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.92e3 + 1.92e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + 1.61e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-784. + 784. i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + 1.12e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.32e4 + 2.32e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + 2.45e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-1.94e4 + 1.94e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.03e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-4.42e4 - 4.42e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + 1.68e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.24e5 - 1.24e5i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (-8.48e4 - 8.48e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.10e5 - 1.10e5i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (2.68e5 - 2.68e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + 1.62e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.85e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 4.82e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-3.04e4 + 3.04e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 4.09e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.45e6T + 8.32e11T^{2} \) |
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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
−14.63925100734968539356364164678, −13.36190341512393598634621335518, −11.63905980399646496650126144988, −11.11665908053405867855228206990, −10.20307662209859109386506843926, −8.021915669972825149383953581480, −7.62627519501836469765364517956, −5.58933194796897844706943916977, −4.30519976206639426562591071336, −2.54250577718861096015521935827,
0.02684472904667779734441910457, 1.65544772846216572561075280688, 4.21942169548339727379611071182, 5.17951910242676845858285741000, 7.16983058315324878755973181742, 8.128967071218323590326001243694, 9.321732572060458321841494601685, 11.07918658987684209782656165057, 12.08510636259386409141987315168, 12.54983632743656812487316069275
