L(s) = 1 | + 10.8·2-s − 9.42·4-s + 125·5-s − 184.·7-s − 1.49e3·8-s + 1.36e3·10-s + 5.12e3·11-s − 3.53e3·13-s − 2.01e3·14-s − 1.50e4·16-s + 4.81e3·17-s − 1.88e4·19-s − 1.17e3·20-s + 5.58e4·22-s + 1.58e4·23-s + 1.56e4·25-s − 3.85e4·26-s + 1.74e3·28-s − 4.01e4·29-s + 8.99e4·31-s + 2.72e4·32-s + 5.24e4·34-s − 2.31e4·35-s − 1.51e5·37-s − 2.05e5·38-s − 1.87e5·40-s + 6.99e5·41-s + ⋯ |
L(s) = 1 | + 0.962·2-s − 0.0736·4-s + 0.447·5-s − 0.203·7-s − 1.03·8-s + 0.430·10-s + 1.16·11-s − 0.446·13-s − 0.196·14-s − 0.920·16-s + 0.237·17-s − 0.631·19-s − 0.0329·20-s + 1.11·22-s + 0.271·23-s + 0.199·25-s − 0.429·26-s + 0.0150·28-s − 0.306·29-s + 0.542·31-s + 0.147·32-s + 0.228·34-s − 0.0911·35-s − 0.491·37-s − 0.607·38-s − 0.462·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.290708888\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.290708888\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
good | 2 | \( 1 - 10.8T + 128T^{2} \) |
| 7 | \( 1 + 184.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.53e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.81e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.88e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.01e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.99e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.51e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.93e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.21e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.45e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.13e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.91e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.30e7T + 8.07e13T^{2} \) |
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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
−9.930171912523552547994726556648, −9.271897114359163994376572074253, −8.367120610062479002240960245398, −6.89926215093496093636599296171, −6.17102200442421818912274563020, −5.20979343119763486039444750128, −4.25548006689187974371689331966, −3.36170946271267805558262545164, −2.16487166989854477321753224421, −0.71855595580175903643557026049,
0.71855595580175903643557026049, 2.16487166989854477321753224421, 3.36170946271267805558262545164, 4.25548006689187974371689331966, 5.20979343119763486039444750128, 6.17102200442421818912274563020, 6.89926215093496093636599296171, 8.367120610062479002240960245398, 9.271897114359163994376572074253, 9.930171912523552547994726556648