| L(s) = 1 | + 5.31·2-s − 27.6·3-s − 3.75·4-s − 23.8·5-s − 147.·6-s − 11.7·7-s − 190.·8-s + 522.·9-s − 126.·10-s + 280.·11-s + 103.·12-s − 835.·13-s − 62.6·14-s + 660.·15-s − 889.·16-s − 1.80e3·17-s + 2.77e3·18-s + 2.35e3·19-s + 89.6·20-s + 325.·21-s + 1.49e3·22-s − 529·23-s + 5.25e3·24-s − 2.55e3·25-s − 4.43e3·26-s − 7.72e3·27-s + 44.2·28-s + ⋯ |
| L(s) = 1 | + 0.939·2-s − 1.77·3-s − 0.117·4-s − 0.427·5-s − 1.66·6-s − 0.0908·7-s − 1.04·8-s + 2.14·9-s − 0.401·10-s + 0.699·11-s + 0.208·12-s − 1.37·13-s − 0.0854·14-s + 0.757·15-s − 0.868·16-s − 1.51·17-s + 2.01·18-s + 1.49·19-s + 0.0501·20-s + 0.161·21-s + 0.657·22-s − 0.208·23-s + 1.86·24-s − 0.817·25-s − 1.28·26-s − 2.03·27-s + 0.0106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 5.31T + 32T^{2} \) |
| 3 | \( 1 + 27.6T + 243T^{2} \) |
| 5 | \( 1 + 23.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 11.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 280.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 835.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.35e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 803.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.87e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.41e4T + 8.58e9T^{2} \) |
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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
−16.19095312274660929696368495159, −14.97426113238234915340638152515, −13.31973432665316582166657565741, −12.07487836008074491234570662027, −11.49103225239884902223432946200, −9.663283700016518715810819299042, −6.93201305117522424877228949487, −5.49408981938770963276211415011, −4.31223549562239653015692499360, 0,
4.31223549562239653015692499360, 5.49408981938770963276211415011, 6.93201305117522424877228949487, 9.663283700016518715810819299042, 11.49103225239884902223432946200, 12.07487836008074491234570662027, 13.31973432665316582166657565741, 14.97426113238234915340638152515, 16.19095312274660929696368495159
