| L(s) = 1 | + 27·3-s − 65.0·5-s − 275.·7-s + 729·9-s + 5.54e3·11-s + 4.95e3·13-s − 1.75e3·15-s − 2.03e4·17-s − 4.91e4·19-s − 7.43e3·21-s − 5.51e4·23-s − 7.38e4·25-s + 1.96e4·27-s + 1.32e5·29-s + 2.48e5·31-s + 1.49e5·33-s + 1.79e4·35-s − 1.58e5·37-s + 1.33e5·39-s + 4.11e5·41-s + 1.65e5·43-s − 4.74e4·45-s − 8.81e5·47-s − 7.47e5·49-s − 5.49e5·51-s + 1.06e6·53-s − 3.60e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.232·5-s − 0.303·7-s + 0.333·9-s + 1.25·11-s + 0.625·13-s − 0.134·15-s − 1.00·17-s − 1.64·19-s − 0.175·21-s − 0.945·23-s − 0.945·25-s + 0.192·27-s + 1.00·29-s + 1.49·31-s + 0.724·33-s + 0.0705·35-s − 0.513·37-s + 0.361·39-s + 0.932·41-s + 0.317·43-s − 0.0775·45-s − 1.23·47-s − 0.908·49-s − 0.579·51-s + 0.986·53-s − 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| good | 5 | \( 1 + 65.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 275.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.95e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.91e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.48e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.11e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.81e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.20e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.23e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.51e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.24e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.42e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.54e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.01e7T + 8.07e13T^{2} \) |
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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
−9.593860837490180174126837042188, −8.687011350709786111221461525956, −8.092890338347545376375516584695, −6.66911692758347002966748265742, −6.22521827866964640522889295986, −4.41024307801941332633572372037, −3.86322255298199291199214484423, −2.53009704566797546566480938077, −1.42834608239971789666392047933, 0,
1.42834608239971789666392047933, 2.53009704566797546566480938077, 3.86322255298199291199214484423, 4.41024307801941332633572372037, 6.22521827866964640522889295986, 6.66911692758347002966748265742, 8.092890338347545376375516584695, 8.687011350709786111221461525956, 9.593860837490180174126837042188
