| L(s) = 1 | − 27i·3-s + 334. i·5-s + 359.·7-s − 729·9-s − 7.65i·11-s + 4.42e3i·13-s + 9.02e3·15-s − 4.95e3·17-s + 1.83e4i·19-s − 9.71e3i·21-s + 3.64e4·23-s − 3.37e4·25-s + 1.96e4i·27-s + 2.85e3i·29-s + 9.01e4·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.19i·5-s + 0.396·7-s − 0.333·9-s − 0.00173i·11-s + 0.558i·13-s + 0.690·15-s − 0.244·17-s + 0.614i·19-s − 0.228i·21-s + 0.624·23-s − 0.431·25-s + 0.192i·27-s + 0.0217i·29-s + 0.543·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.286108419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286108419\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 - 334. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 359.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.65iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 4.42e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 4.95e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.83e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 3.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.85e3iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 9.01e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.81e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 1.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.44e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 6.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.91e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.72e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.64e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.21e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.85e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.09e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.69e5T + 8.07e13T^{2} \) |
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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
−10.67561348664216070854670186835, −9.650450662576357749523364499451, −8.512648811995661779988147555741, −7.56530205438999820484297893747, −6.78075130341190919823181006688, −6.02269940430053987327433752202, −4.67720915196792102503390390253, −3.35655948955778841451148416994, −2.38640057063543077218116224258, −1.28904448936093639153097395334,
0.27288863584751004556411276886, 1.30498134153901688934995939362, 2.74208643549759955293272895588, 4.10148804270459475689066053916, 4.92901639711799479414250386265, 5.63704396178168713020729795184, 7.05081610848845057774030301101, 8.281593446739101368415144341450, 8.826610208152061280902143854339, 9.721503346141663665176883859086
