| L(s) = 1 | + 455. i·5-s + 743.·7-s + 5.47e3i·11-s + 6.21e3i·13-s + 2.63e4·17-s + 2.33e4i·19-s + 5.09e4·23-s − 1.29e5·25-s + 1.85e5i·29-s + 1.84e5·31-s + 3.38e5i·35-s − 2.35e5i·37-s + 5.39e5·41-s − 3.04e5i·43-s + 9.23e5·47-s + ⋯ |
| L(s) = 1 | + 1.62i·5-s + 0.819·7-s + 1.24i·11-s + 0.785i·13-s + 1.29·17-s + 0.780i·19-s + 0.873·23-s − 1.65·25-s + 1.40i·29-s + 1.11·31-s + 1.33i·35-s − 0.764i·37-s + 1.22·41-s − 0.583i·43-s + 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.733i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.627236238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627236238\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 455. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 743.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.47e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 6.21e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.33e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 5.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.85e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.35e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.04e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.21e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.07e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.76e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.56e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.93e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.77e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 9.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.83e6T + 8.07e13T^{2} \) |
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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
−10.79721863688197843709257609877, −10.22448471581779602119351156420, −9.169860252580920246320507967860, −7.68890927473631944133542999315, −7.21902300948919655531943986361, −6.16565854665079007635548861107, −4.87759915209171838169364811770, −3.66234759557034659631229227541, −2.50963225255894919501854122161, −1.44120033024846580027499053467,
0.69992143486521365565122376438, 1.11678681938188011168177557638, 2.85287328542036383490641577524, 4.30430550542727771535752452141, 5.20757934817652287627270685594, 5.93253392075317279762032327068, 7.72362705946663597164464781712, 8.331489161671130858100146681139, 9.078519178225326744942936593810, 10.18537970251716118881134120983
