| L(s) = 1 | + 15.1·2-s − 27·3-s + 102.·4-s + 149.·5-s − 409.·6-s − 45.6·7-s − 389.·8-s + 729·9-s + 2.26e3·10-s + 6.85e3·11-s − 2.76e3·12-s + 1.22e4·13-s − 693.·14-s − 4.03e3·15-s − 1.90e4·16-s + 7.81e3·17-s + 1.10e4·18-s + 1.70e4·19-s + 1.52e4·20-s + 1.23e3·21-s + 1.04e5·22-s − 1.21e4·23-s + 1.05e4·24-s − 5.58e4·25-s + 1.86e5·26-s − 1.96e4·27-s − 4.67e3·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.577·3-s + 0.799·4-s + 0.534·5-s − 0.774·6-s − 0.0503·7-s − 0.269·8-s + 0.333·9-s + 0.716·10-s + 1.55·11-s − 0.461·12-s + 1.55·13-s − 0.0675·14-s − 0.308·15-s − 1.16·16-s + 0.385·17-s + 0.447·18-s + 0.569·19-s + 0.427·20-s + 0.0290·21-s + 2.08·22-s − 0.208·23-s + 0.155·24-s − 0.714·25-s + 2.07·26-s − 0.192·27-s − 0.0402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.790698524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790698524\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 15.1T + 128T^{2} \) |
| 5 | \( 1 - 149.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 45.6T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.85e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.81e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.70e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.17e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.59e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.53e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.39e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.66e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.79e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.33e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.60e6T + 8.07e13T^{2} \) |
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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
−13.62416067333235908403083430507, −12.12279146752430875068571626956, −11.61790280669147846685264462905, −10.08974859959501420370891261012, −8.715662181214572960896033716803, −6.53758940231096112106700906777, −5.98214005743766049256817708184, −4.56522658774148989594667180495, −3.37674319901265919732896229833, −1.28360327430799544793536801526,
1.28360327430799544793536801526, 3.37674319901265919732896229833, 4.56522658774148989594667180495, 5.98214005743766049256817708184, 6.53758940231096112106700906777, 8.715662181214572960896033716803, 10.08974859959501420370891261012, 11.61790280669147846685264462905, 12.12279146752430875068571626956, 13.62416067333235908403083430507
