| L(s) = 1 | + 90.0·2-s + 729·3-s − 84.2·4-s − 4.01e4·5-s + 6.56e4·6-s − 3.37e5·7-s − 7.45e5·8-s + 5.31e5·9-s − 3.61e6·10-s − 7.78e6·11-s − 6.14e4·12-s + 1.67e6·13-s − 3.04e7·14-s − 2.92e7·15-s − 6.64e7·16-s − 1.97e8·17-s + 4.78e7·18-s + 1.04e8·19-s + 3.38e6·20-s − 2.46e8·21-s − 7.00e8·22-s − 6.39e8·23-s − 5.43e8·24-s + 3.94e8·25-s + 1.50e8·26-s + 3.87e8·27-s + 2.84e7·28-s + ⋯ |
| L(s) = 1 | + 0.994·2-s + 0.577·3-s − 0.0102·4-s − 1.15·5-s + 0.574·6-s − 1.08·7-s − 1.00·8-s + 0.333·9-s − 1.14·10-s − 1.32·11-s − 0.00593·12-s + 0.0959·13-s − 1.08·14-s − 0.664·15-s − 0.989·16-s − 1.98·17-s + 0.331·18-s + 0.509·19-s + 0.0118·20-s − 0.626·21-s − 1.31·22-s − 0.901·23-s − 0.580·24-s + 0.322·25-s + 0.0954·26-s + 0.192·27-s + 0.0111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.1528438884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1528438884\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 90.0T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.01e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.37e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.78e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.67e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.97e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.04e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.39e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.05e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.85e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.15e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.18e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.04e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 2.14e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.32e10T + 2.60e22T^{2} \) |
| 61 | \( 1 + 4.78e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.27e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.28e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.47e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 6.13e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.96e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 3.68e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.38e12T + 6.73e25T^{2} \) |
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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
−10.38921586897105111597103920462, −9.215411011877578250326959746389, −8.338696355818733497647095093611, −7.26620772275862320261284587229, −6.18372901404599231978101275914, −4.90208076000056504092940551568, −3.99308446676035837595816989917, −3.25651673704676481421582166054, −2.34827879600838760253496798773, −0.12674835243356723448119483828,
0.12674835243356723448119483828, 2.34827879600838760253496798773, 3.25651673704676481421582166054, 3.99308446676035837595816989917, 4.90208076000056504092940551568, 6.18372901404599231978101275914, 7.26620772275862320261284587229, 8.338696355818733497647095093611, 9.215411011877578250326959746389, 10.38921586897105111597103920462
