| L(s) = 1 | + 2.02e7·2-s + 2.71e14·4-s + 3.11e16·5-s − 1.26e20·7-s + 2.64e21·8-s + 6.32e23·10-s + 1.78e23·11-s − 1.12e26·13-s − 2.56e27·14-s + 1.55e28·16-s − 4.31e28·17-s − 7.19e29·19-s + 8.45e30·20-s + 3.62e30·22-s − 8.88e30·23-s + 2.60e32·25-s − 2.28e33·26-s − 3.43e34·28-s − 2.84e34·29-s + 1.11e35·31-s − 5.62e34·32-s − 8.75e35·34-s − 3.94e36·35-s − 9.87e36·37-s − 1.46e37·38-s + 8.25e37·40-s + 3.56e37·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.92·4-s + 1.16·5-s − 1.74·7-s + 1.58·8-s + 2.00·10-s + 0.0601·11-s − 0.746·13-s − 2.99·14-s + 0.787·16-s − 0.523·17-s − 0.640·19-s + 2.25·20-s + 0.102·22-s − 0.0886·23-s + 0.367·25-s − 1.27·26-s − 3.36·28-s − 1.22·29-s + 1.00·31-s − 0.239·32-s − 0.895·34-s − 2.04·35-s − 1.38·37-s − 1.09·38-s + 1.85·40-s + 0.447·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{49}{2})\) |
|
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 | \( 1 - 2.02e7T + 1.40e14T^{2} \) |
| 5 | \( 1 - 3.11e16T + 7.10e32T^{2} \) |
| 7 | \( 1 + 1.26e20T + 5.24e39T^{2} \) |
| 11 | \( 1 - 1.78e23T + 8.81e48T^{2} \) |
| 13 | \( 1 + 1.12e26T + 2.26e52T^{2} \) |
| 17 | \( 1 + 4.31e28T + 6.77e57T^{2} \) |
| 19 | \( 1 + 7.19e29T + 1.26e60T^{2} \) |
| 23 | \( 1 + 8.88e30T + 1.00e64T^{2} \) |
| 29 | \( 1 + 2.84e34T + 5.40e68T^{2} \) |
| 31 | \( 1 - 1.11e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 9.87e36T + 5.07e73T^{2} \) |
| 41 | \( 1 - 3.56e37T + 6.32e75T^{2} \) |
| 43 | \( 1 + 3.37e38T + 5.92e76T^{2} \) |
| 47 | \( 1 - 1.22e39T + 3.87e78T^{2} \) |
| 53 | \( 1 - 1.08e39T + 1.09e81T^{2} \) |
| 59 | \( 1 - 1.45e40T + 1.69e83T^{2} \) |
| 61 | \( 1 - 4.79e41T + 8.13e83T^{2} \) |
| 67 | \( 1 - 3.35e42T + 6.69e85T^{2} \) |
| 71 | \( 1 + 9.20e42T + 1.02e87T^{2} \) |
| 73 | \( 1 + 5.22e43T + 3.76e87T^{2} \) |
| 79 | \( 1 + 7.68e43T + 1.54e89T^{2} \) |
| 83 | \( 1 - 1.71e45T + 1.57e90T^{2} \) |
| 89 | \( 1 - 9.70e45T + 4.18e91T^{2} \) |
| 97 | \( 1 - 5.35e46T + 2.38e93T^{2} \) |
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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
−12.10664786284634376221201743360, −10.38935411916911604265203999006, −9.323785684141034161967642706586, −6.87997770760871855238877251470, −6.21252169421993395507075390963, −5.26260994389718562003197785204, −3.89513201925732124911927413341, −2.82767822461899807369969039060, −1.99842236289298093432167142576, 0,
1.99842236289298093432167142576, 2.82767822461899807369969039060, 3.89513201925732124911927413341, 5.26260994389718562003197785204, 6.21252169421993395507075390963, 6.87997770760871855238877251470, 9.323785684141034161967642706586, 10.38935411916911604265203999006, 12.10664786284634376221201743360
