L(s) = 1 | + 21.8·2-s − 27·3-s + 347.·4-s + 424.·5-s − 588.·6-s + 1.19e3·7-s + 4.79e3·8-s + 729·9-s + 9.26e3·10-s − 3.09e3·11-s − 9.39e3·12-s + 1.04e3·13-s + 2.60e4·14-s − 1.14e4·15-s + 6.00e4·16-s − 1.66e4·17-s + 1.59e4·18-s + 2.55e4·19-s + 1.47e5·20-s − 3.22e4·21-s − 6.74e4·22-s − 9.87e4·23-s − 1.29e5·24-s + 1.02e5·25-s + 2.27e4·26-s − 1.96e4·27-s + 4.14e5·28-s + ⋯ |
L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.52·5-s − 1.11·6-s + 1.31·7-s + 3.31·8-s + 0.333·9-s + 2.93·10-s − 0.700·11-s − 1.56·12-s + 0.131·13-s + 2.53·14-s − 0.877·15-s + 3.66·16-s − 0.820·17-s + 0.642·18-s + 0.855·19-s + 4.13·20-s − 0.758·21-s − 1.35·22-s − 1.69·23-s − 1.91·24-s + 1.31·25-s + 0.254·26-s − 0.192·27-s + 3.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(9.199812643\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.199812643\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 2 | \( 1 - 21.8T + 128T^{2} \) |
| 5 | \( 1 - 424.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.04e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.87e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.17e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 7.34e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.17e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.09e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.13e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.73e6T + 8.07e13T^{2} \) |
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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
−11.47121142296658693599483848291, −10.93739922461963767146489576749, −9.846278031941274919398867895475, −7.85184122277972456628283315301, −6.65995132698684885293295818356, −5.49009809986635294350764448399, −5.33392391850219914261465509094, −4.02781204427356416404371460246, −2.27603174719015657520765618953, −1.66709054492680548484409165125,
1.66709054492680548484409165125, 2.27603174719015657520765618953, 4.02781204427356416404371460246, 5.33392391850219914261465509094, 5.49009809986635294350764448399, 6.65995132698684885293295818356, 7.85184122277972456628283315301, 9.846278031941274919398867895475, 10.93739922461963767146489576749, 11.47121142296658693599483848291