L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 45.2·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s + 361.·10-s + 3.72e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 1.22e3·15-s + 4.09e3·16-s + 2.43e4·17-s + 5.83e3·18-s + 1.61e4·19-s + 2.89e3·20-s + 9.26e3·21-s + 2.97e4·22-s − 1.97e4·23-s − 1.38e4·24-s − 7.60e4·25-s + 1.75e4·26-s − 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.161·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.114·10-s + 0.843·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0934·15-s + 0.250·16-s + 1.20·17-s + 0.235·18-s + 0.540·19-s + 0.0809·20-s + 0.218·21-s + 0.596·22-s − 0.338·23-s − 0.204·24-s − 0.973·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.414922403\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.414922403\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 5 | \( 1 - 45.2T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.72e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.43e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.61e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.26e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.70e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.26e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.04e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.64e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.35e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.09e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.10e7T + 8.07e13T^{2} \) |
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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
−9.932202402290852122596971926566, −8.837272039671269309688465674699, −7.62110251342880651385298586155, −6.71930858661035359583796375181, −5.93212101908211325554215489421, −5.17734509244691403455424725103, −4.00645866369269924869648568940, −3.21655153845135260281412088250, −1.78540108812861357021929440290, −0.76295838478442390323343570313,
0.76295838478442390323343570313, 1.78540108812861357021929440290, 3.21655153845135260281412088250, 4.00645866369269924869648568940, 5.17734509244691403455424725103, 5.93212101908211325554215489421, 6.71930858661035359583796375181, 7.62110251342880651385298586155, 8.837272039671269309688465674699, 9.932202402290852122596971926566