| L(s) = 1 | + 60.2·2-s + 729·3-s − 4.56e3·4-s − 2.87e3·5-s + 4.39e4·6-s + 1.93e4·7-s − 7.68e5·8-s + 5.31e5·9-s − 1.73e5·10-s + 1.70e6·11-s − 3.32e6·12-s + 4.41e6·13-s + 1.16e6·14-s − 2.09e6·15-s − 8.88e6·16-s + 4.11e7·17-s + 3.20e7·18-s − 1.87e8·19-s + 1.31e7·20-s + 1.40e7·21-s + 1.02e8·22-s + 8.00e8·23-s − 5.60e8·24-s − 1.21e9·25-s + 2.65e8·26-s + 3.87e8·27-s − 8.81e7·28-s + ⋯ |
| L(s) = 1 | + 0.665·2-s + 0.577·3-s − 0.557·4-s − 0.0824·5-s + 0.384·6-s + 0.0620·7-s − 1.03·8-s + 0.333·9-s − 0.0548·10-s + 0.290·11-s − 0.321·12-s + 0.253·13-s + 0.0412·14-s − 0.0475·15-s − 0.132·16-s + 0.413·17-s + 0.221·18-s − 0.912·19-s + 0.0459·20-s + 0.0358·21-s + 0.193·22-s + 1.12·23-s − 0.598·24-s − 0.993·25-s + 0.168·26-s + 0.192·27-s − 0.0345·28-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 60.2T + 8.19e3T^{2} \) |
| 5 | \( 1 + 2.87e3T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.93e4T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.70e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 4.41e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 4.11e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.87e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 8.00e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.17e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.99e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 7.40e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.23e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.18e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.69e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 4.59e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.96e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.13e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 9.45e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.00e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.28e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.14e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.88e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.59e13T + 6.73e25T^{2} \) |
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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
−9.654346447412160247534004744711, −8.859621430546987088750315263936, −7.990213776872496778492295459370, −6.68437387668168479409799313729, −5.57256478789645809839189276590, −4.47830623511929086979355423361, −3.67210410669868542090380334305, −2.68027601518753127944514337671, −1.29434792181870696870447451618, 0,
1.29434792181870696870447451618, 2.68027601518753127944514337671, 3.67210410669868542090380334305, 4.47830623511929086979355423361, 5.57256478789645809839189276590, 6.68437387668168479409799313729, 7.990213776872496778492295459370, 8.859621430546987088750315263936, 9.654346447412160247534004744711
