| L(s) = 1 | − 35.4·2-s + 64.4·3-s + 747.·4-s + 1.77e3·5-s − 2.28e3·6-s − 1.60e3·7-s − 8.36e3·8-s − 1.55e4·9-s − 6.31e4·10-s − 3.81e4·11-s + 4.81e4·12-s − 1.16e5·13-s + 5.68e4·14-s + 1.14e5·15-s − 8.59e4·16-s + 5.51e5·18-s + 8.50e5·19-s + 1.33e6·20-s − 1.03e5·21-s + 1.35e6·22-s + 1.05e6·23-s − 5.39e5·24-s + 1.21e6·25-s + 4.13e6·26-s − 2.26e6·27-s − 1.19e6·28-s − 1.44e6·29-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.459·3-s + 1.46·4-s + 1.27·5-s − 0.720·6-s − 0.252·7-s − 0.722·8-s − 0.789·9-s − 1.99·10-s − 0.786·11-s + 0.670·12-s − 1.13·13-s + 0.395·14-s + 0.584·15-s − 0.327·16-s + 1.23·18-s + 1.49·19-s + 1.85·20-s − 0.115·21-s + 1.23·22-s + 0.785·23-s − 0.331·24-s + 0.621·25-s + 1.77·26-s − 0.821·27-s − 0.368·28-s − 0.379·29-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 | \( 1 + 35.4T + 512T^{2} \) |
| 3 | \( 1 - 64.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.77e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.60e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.81e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.16e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.05e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.51e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.30e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.73e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.62e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.35e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.85e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.05e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.21e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.34e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.88e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.76e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.87e8T + 7.60e17T^{2} \) |
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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
−9.472290252264847720092687417527, −9.183964225536682292004552780925, −7.960448740567222999113063012794, −7.28640441742441167407221632659, −6.02648181652324292586434668519, −5.05100269671711646739478597376, −2.87488070142401114884026295175, −2.33559265205905842456688350955, −1.13631391351414857451093452455, 0,
1.13631391351414857451093452455, 2.33559265205905842456688350955, 2.87488070142401114884026295175, 5.05100269671711646739478597376, 6.02648181652324292586434668519, 7.28640441742441167407221632659, 7.960448740567222999113063012794, 9.183964225536682292004552780925, 9.472290252264847720092687417527
