| L(s) = 1 | + 37.7·2-s − 226.·3-s + 910.·4-s − 2.06e3·5-s − 8.53e3·6-s − 6.87e3·7-s + 1.50e4·8-s + 3.15e4·9-s − 7.77e4·10-s + 6.14e4·11-s − 2.06e5·12-s + 1.45e5·13-s − 2.59e5·14-s + 4.66e5·15-s + 1.00e5·16-s + 1.18e6·18-s − 6.68e5·19-s − 1.87e6·20-s + 1.55e6·21-s + 2.31e6·22-s − 4.70e5·23-s − 3.40e6·24-s + 2.29e6·25-s + 5.47e6·26-s − 2.67e6·27-s − 6.25e6·28-s + 5.12e6·29-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 1.61·3-s + 1.77·4-s − 1.47·5-s − 2.68·6-s − 1.08·7-s + 1.29·8-s + 1.60·9-s − 2.45·10-s + 1.26·11-s − 2.86·12-s + 1.41·13-s − 1.80·14-s + 2.37·15-s + 0.385·16-s + 2.66·18-s − 1.17·19-s − 2.62·20-s + 1.74·21-s + 2.10·22-s − 0.350·23-s − 2.09·24-s + 1.17·25-s + 2.35·26-s − 0.968·27-s − 1.92·28-s + 1.34·29-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 | \( 1 - 37.7T + 512T^{2} \) |
| 3 | \( 1 + 226.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.06e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.14e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.45e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 6.68e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 4.70e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.72e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.09e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 7.03e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.76e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.22e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.21e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.82e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.28e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.38e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.81e8T + 7.60e17T^{2} \) |
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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
−10.40441299073455670964540181985, −8.697246675152642786504851415646, −7.10990631233084100538946051610, −6.27414373898285200571766864941, −6.04031229898044937060795900961, −4.37952879972107388834441917181, −4.17210356925158909447156533384, −3.11961652784173738921832112071, −1.08669818420129574852231086414, 0,
1.08669818420129574852231086414, 3.11961652784173738921832112071, 4.17210356925158909447156533384, 4.37952879972107388834441917181, 6.04031229898044937060795900961, 6.27414373898285200571766864941, 7.10990631233084100538946051610, 8.697246675152642786504851415646, 10.40441299073455670964540181985
