L(s) = 1 | − 14.7·2-s − 27·3-s + 91.0·4-s − 296.·5-s + 399.·6-s − 1.41e3·7-s + 547.·8-s + 729·9-s + 4.38e3·10-s − 7.53e3·11-s − 2.45e3·12-s + 7.06e3·13-s + 2.08e4·14-s + 8.00e3·15-s − 1.97e4·16-s + 1.48e4·17-s − 1.07e4·18-s − 2.40e4·19-s − 2.69e4·20-s + 3.80e4·21-s + 1.11e5·22-s + 4.15e4·23-s − 1.47e4·24-s + 9.80e3·25-s − 1.04e5·26-s − 1.96e4·27-s − 1.28e5·28-s + ⋯ |
L(s) = 1 | − 1.30·2-s − 0.577·3-s + 0.711·4-s − 1.06·5-s + 0.755·6-s − 1.55·7-s + 0.378·8-s + 0.333·9-s + 1.38·10-s − 1.70·11-s − 0.410·12-s + 0.892·13-s + 2.03·14-s + 0.612·15-s − 1.20·16-s + 0.732·17-s − 0.436·18-s − 0.803·19-s − 0.754·20-s + 0.897·21-s + 2.23·22-s + 0.712·23-s − 0.218·24-s + 0.125·25-s − 1.16·26-s − 0.192·27-s − 1.10·28-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
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 14.7T + 128T^{2} \) |
| 5 | \( 1 + 296.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.41e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.53e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.06e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.57e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.16e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.21e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.88e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.24e6T + 8.07e13T^{2} \) |
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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
−10.55961593722697086216410974682, −10.06415182247132923272336029678, −8.831985853887272903406410170785, −7.890268100049190470794606520326, −7.05448766698217224135325845910, −5.81780227344300270267591646152, −4.21180787431480539445966581307, −2.82248641663990520093085642815, −0.77668908563956621872924272170, 0,
0.77668908563956621872924272170, 2.82248641663990520093085642815, 4.21180787431480539445966581307, 5.81780227344300270267591646152, 7.05448766698217224135325845910, 7.890268100049190470794606520326, 8.831985853887272903406410170785, 10.06415182247132923272336029678, 10.55961593722697086216410974682