| L(s) = 1 | + 27·3-s + 118.·5-s + 343·7-s + 729·9-s + 3.52e3·11-s − 5.25e3·13-s + 3.21e3·15-s − 3.70e4·17-s − 5.16e3·19-s + 9.26e3·21-s − 3.45e4·23-s − 6.39e4·25-s + 1.96e4·27-s + 6.38e3·29-s − 1.55e5·31-s + 9.50e4·33-s + 4.08e4·35-s + 1.87e4·37-s − 1.41e5·39-s + 2.61e5·41-s + 1.34e5·43-s + 8.67e4·45-s − 1.08e6·47-s + 1.17e5·49-s − 9.99e5·51-s + 4.04e5·53-s + 4.18e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.425·5-s + 0.377·7-s + 0.333·9-s + 0.797·11-s − 0.663·13-s + 0.245·15-s − 1.82·17-s − 0.172·19-s + 0.218·21-s − 0.592·23-s − 0.818·25-s + 0.192·27-s + 0.0485·29-s − 0.939·31-s + 0.460·33-s + 0.160·35-s + 0.0609·37-s − 0.383·39-s + 0.592·41-s + 0.258·43-s + 0.141·45-s − 1.52·47-s + 0.142·49-s − 1.05·51-s + 0.373·53-s + 0.339·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 - 118.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.25e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.16e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.38e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.55e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.87e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.61e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.04e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.06e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−9.663011799811670967490421113835, −9.056935480822121398349910696761, −8.081497639341731725676751717590, −7.03266049035976927905546805240, −6.10557969872377582538562061862, −4.76194510222791804775413367295, −3.86191499242577659594568848115, −2.40927697006983942398718360252, −1.64260146839291208649482400482, 0,
1.64260146839291208649482400482, 2.40927697006983942398718360252, 3.86191499242577659594568848115, 4.76194510222791804775413367295, 6.10557969872377582538562061862, 7.03266049035976927905546805240, 8.081497639341731725676751717590, 9.056935480822121398349910696761, 9.663011799811670967490421113835
