| L(s) = 1 | + (11.0 + 11.0i)3-s + (26.6 + 26.6i)5-s + 403.·7-s + 242. i·9-s + (152. − 152. i)11-s + (1.72e3 − 1.72e3i)13-s + 587. i·15-s + 110.·17-s + (3.29e3 + 3.29e3i)19-s + (4.45e3 + 4.45e3i)21-s + 4.14e3·23-s − 1.42e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (2.62e4 − 2.62e4i)29-s − 1.55e4i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (0.213 + 0.213i)5-s + 1.17·7-s + 0.333i·9-s + (0.114 − 0.114i)11-s + (0.786 − 0.786i)13-s + 0.174i·15-s + 0.0225·17-s + (0.480 + 0.480i)19-s + (0.480 + 0.480i)21-s + 0.340·23-s − 0.909i·25-s + (−0.136 + 0.136i)27-s + (1.07 − 1.07i)29-s − 0.520i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0414i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 - 0.0414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.477613113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.477613113\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 - 11.0i)T \) |
| good | 5 | \( 1 + (-26.6 - 26.6i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 403.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-152. + 152. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (-1.72e3 + 1.72e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 110.T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-3.29e3 - 3.29e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 4.14e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-2.62e4 + 2.62e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 1.55e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (4.18e4 + 4.18e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.35e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-2.55e4 + 2.55e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 3.65e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.15e5 + 1.15e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-7.10e4 + 7.10e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-9.63e4 + 9.63e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (2.66e5 + 2.66e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 4.16e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 2.02e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 8.32e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (1.57e5 + 1.57e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 9.85e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.46e6T + 8.32e11T^{2} \) |
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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
−10.37486588471410108288354183447, −9.472955471339092593425217021508, −8.238692309577250611676240082941, −7.978123981491768859377686558480, −6.48183094604576929097958990268, −5.41360571858735684414395626667, −4.43278761146252988233650415736, −3.30302373680628903483661832492, −2.10110071948940963197817756757, −0.865596687311613403467267731351,
1.11007567424314711360898467807, 1.79228958726859463129791829508, 3.19594014036872925508631951185, 4.49287988613104299453931004260, 5.42083680204376215060451514069, 6.72075658641343545535772527751, 7.52391713635722055839055077906, 8.669068844868035996823684137313, 9.032394692645419245830479225797, 10.41423022385601243061515841401
