| L(s) = 1 | − 9.15·2-s + 9.09·3-s + 51.7·4-s + 3.86·5-s − 83.2·6-s − 226.·7-s − 180.·8-s − 160.·9-s − 35.3·10-s + 18.3·11-s + 470.·12-s − 711.·13-s + 2.07e3·14-s + 35.1·15-s − 3.60·16-s + 165.·17-s + 1.46e3·18-s + 213.·19-s + 199.·20-s − 2.05e3·21-s − 167.·22-s − 529·23-s − 1.64e3·24-s − 3.11e3·25-s + 6.50e3·26-s − 3.66e3·27-s − 1.17e4·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.583·3-s + 1.61·4-s + 0.0691·5-s − 0.943·6-s − 1.74·7-s − 0.997·8-s − 0.659·9-s − 0.111·10-s + 0.0456·11-s + 0.943·12-s − 1.16·13-s + 2.82·14-s + 0.0403·15-s − 0.00352·16-s + 0.138·17-s + 1.06·18-s + 0.135·19-s + 0.111·20-s − 1.01·21-s − 0.0738·22-s − 0.208·23-s − 0.581·24-s − 0.995·25-s + 1.88·26-s − 0.968·27-s − 2.82·28-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 | \( 1 + 9.15T + 32T^{2} \) |
| 3 | \( 1 - 9.09T + 243T^{2} \) |
| 5 | \( 1 - 3.86T + 3.12e3T^{2} \) |
| 7 | \( 1 + 226.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 18.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 711.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 165.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 213.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 5.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.91e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e5T + 8.58e9T^{2} \) |
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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
−16.52216286400100048398235069591, −15.37947389080488218696688756904, −13.64776901001646144077118082941, −11.96117264130566752377549293114, −10.07161753068757523632361854977, −9.426206077462713453649794813770, −8.054824913865107838436730598268, −6.54469224639957625151710339711, −2.77232553239567056013537377254, 0,
2.77232553239567056013537377254, 6.54469224639957625151710339711, 8.054824913865107838436730598268, 9.426206077462713453649794813770, 10.07161753068757523632361854977, 11.96117264130566752377549293114, 13.64776901001646144077118082941, 15.37947389080488218696688756904, 16.52216286400100048398235069591
