| L(s) = 1 | + 28.8·2-s − 253.·3-s + 319.·4-s + 1.60e3·5-s − 7.31e3·6-s − 6.96e3·7-s − 5.55e3·8-s + 4.46e4·9-s + 4.62e4·10-s + 6.33e4·11-s − 8.10e4·12-s − 1.14e5·13-s − 2.00e5·14-s − 4.06e5·15-s − 3.23e5·16-s + 1.28e6·18-s + 5.04e4·19-s + 5.11e5·20-s + 1.76e6·21-s + 1.82e6·22-s + 1.13e6·23-s + 1.40e6·24-s + 6.15e5·25-s − 3.30e6·26-s − 6.34e6·27-s − 2.22e6·28-s + 2.03e6·29-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 1.80·3-s + 0.623·4-s + 1.14·5-s − 2.30·6-s − 1.09·7-s − 0.479·8-s + 2.26·9-s + 1.46·10-s + 1.30·11-s − 1.12·12-s − 1.11·13-s − 1.39·14-s − 2.07·15-s − 1.23·16-s + 2.89·18-s + 0.0888·19-s + 0.715·20-s + 1.98·21-s + 1.66·22-s + 0.848·23-s + 0.867·24-s + 0.314·25-s − 1.41·26-s − 2.29·27-s − 0.683·28-s + 0.535·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 28.8T + 512T^{2} \) |
| 3 | \( 1 + 253.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.60e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.96e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.33e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.14e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 5.04e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.03e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.84e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.96e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.09e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.43e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.61e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.17e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.52e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.20e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.83e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.28e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.66e8T + 7.60e17T^{2} \) |
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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
−9.757262158543382944713496556134, −9.436470245155159296232425736004, −6.92971500094072007137222608192, −6.40179816041379946115526276188, −5.78760348995724540739655040999, −4.96191065922958685876810445799, −4.07506849362680188644842918576, −2.66687929208234985323762237771, −1.16926488120397675191865273063, 0,
1.16926488120397675191865273063, 2.66687929208234985323762237771, 4.07506849362680188644842918576, 4.96191065922958685876810445799, 5.78760348995724540739655040999, 6.40179816041379946115526276188, 6.92971500094072007137222608192, 9.436470245155159296232425736004, 9.757262158543382944713496556134
