| L(s) = 1 | − 2.77e3·2-s + 5.61e6·4-s − 3.15e7·5-s + 5.45e8·7-s − 9.75e9·8-s + 8.75e10·10-s + 9.88e10·11-s − 4.22e11·13-s − 1.51e12·14-s + 1.53e13·16-s + 8.22e11·17-s − 2.94e10·19-s − 1.76e14·20-s − 2.74e14·22-s − 1.03e13·23-s + 5.17e14·25-s + 1.17e15·26-s + 3.05e15·28-s − 2.95e15·29-s + 4.38e15·31-s − 2.20e16·32-s − 2.28e15·34-s − 1.71e16·35-s + 4.72e16·37-s + 8.16e13·38-s + 3.07e17·40-s + 2.86e15·41-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.67·4-s − 1.44·5-s + 0.729·7-s − 3.21·8-s + 2.76·10-s + 1.14·11-s − 0.849·13-s − 1.39·14-s + 3.48·16-s + 0.0989·17-s − 0.00110·19-s − 3.86·20-s − 2.20·22-s − 0.0522·23-s + 1.08·25-s + 1.62·26-s + 1.95·28-s − 1.30·29-s + 0.961·31-s − 3.46·32-s − 0.189·34-s − 1.05·35-s + 1.61·37-s + 0.00211·38-s + 4.63·40-s + 0.0333·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.77e3T + 2.09e6T^{2} \) |
| 5 | \( 1 + 3.15e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 5.45e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 9.88e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 4.22e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.22e11T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.94e10T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.03e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.95e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.38e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.72e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.86e15T + 7.38e33T^{2} \) |
| 43 | \( 1 - 8.15e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.66e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.19e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.70e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.75e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.30e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.48e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.60e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.72e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.11e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.67e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.61e20T + 5.27e41T^{2} \) |
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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
−9.748061964005811045105851721414, −8.869129377613549399585111413145, −7.84548452769371505353916504214, −7.46126000713867130640572630828, −6.27921586752787892810622170368, −4.42504753123432752008706547559, −3.11559016890610716162001997933, −1.82920487330190709269865925739, −0.880415163710535721454422587404, 0,
0.880415163710535721454422587404, 1.82920487330190709269865925739, 3.11559016890610716162001997933, 4.42504753123432752008706547559, 6.27921586752787892810622170368, 7.46126000713867130640572630828, 7.84548452769371505353916504214, 8.869129377613549399585111413145, 9.748061964005811045105851721414
