| L(s) = 1 | + 81·3-s − 857.·5-s − 1.12e4·7-s + 6.56e3·9-s + 3.72e4·11-s + 1.15e5·13-s − 6.94e4·15-s + 2.19e5·17-s − 1.71e5·19-s − 9.08e5·21-s − 1.41e6·23-s − 1.21e6·25-s + 5.31e5·27-s + 5.64e6·29-s − 9.34e6·31-s + 3.02e6·33-s + 9.62e6·35-s − 2.64e6·37-s + 9.34e6·39-s + 1.89e7·41-s + 3.40e7·43-s − 5.62e6·45-s − 2.32e6·47-s + 8.55e7·49-s + 1.78e7·51-s + 1.38e7·53-s − 3.19e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.613·5-s − 1.76·7-s + 0.333·9-s + 0.767·11-s + 1.12·13-s − 0.354·15-s + 0.638·17-s − 0.301·19-s − 1.01·21-s − 1.05·23-s − 0.623·25-s + 0.192·27-s + 1.48·29-s − 1.81·31-s + 0.443·33-s + 1.08·35-s − 0.232·37-s + 0.647·39-s + 1.04·41-s + 1.51·43-s − 0.204·45-s − 0.0694·47-s + 2.12·49-s + 0.368·51-s + 0.241·53-s − 0.471·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| good | 5 | \( 1 + 857.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.12e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.72e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.15e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.19e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.71e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.64e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.34e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.64e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.89e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.40e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.32e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.38e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.00e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.17e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.99e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.44e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.20e8T + 7.60e17T^{2} \) |
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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
−9.341356006035380346066254153925, −8.543834692126449721164834687278, −7.51557101529750483523596712409, −6.54384721045711276782128292284, −5.82538011664825526601223905995, −3.89563283653776124238634081292, −3.72617489850809642039729800967, −2.54657408127349358583183957124, −1.10794941666152551232830507883, 0,
1.10794941666152551232830507883, 2.54657408127349358583183957124, 3.72617489850809642039729800967, 3.89563283653776124238634081292, 5.82538011664825526601223905995, 6.54384721045711276782128292284, 7.51557101529750483523596712409, 8.543834692126449721164834687278, 9.341356006035380346066254153925
