| L(s) = 1 | − 2.83·5-s − 17.1·7-s − 48.7·11-s − 64.6·13-s − 118.·17-s − 99.4·19-s − 57.9·23-s − 116.·25-s − 27.3·29-s + 58.0·31-s + 48.5·35-s + 133.·37-s + 195.·41-s + 351.·43-s + 568.·47-s − 50.5·49-s − 208.·53-s + 138.·55-s − 810.·59-s − 583.·61-s + 183.·65-s − 22.5·67-s + 218.·71-s + 273.·73-s + 833.·77-s + 830.·79-s − 650.·83-s + ⋯ |
| L(s) = 1 | − 0.253·5-s − 0.923·7-s − 1.33·11-s − 1.37·13-s − 1.69·17-s − 1.20·19-s − 0.525·23-s − 0.935·25-s − 0.175·29-s + 0.336·31-s + 0.234·35-s + 0.592·37-s + 0.743·41-s + 1.24·43-s + 1.76·47-s − 0.147·49-s − 0.539·53-s + 0.338·55-s − 1.78·59-s − 1.22·61-s + 0.349·65-s − 0.0411·67-s + 0.365·71-s + 0.438·73-s + 1.23·77-s + 1.18·79-s − 0.860·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1715761714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1715761714\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.83T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 + 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 810.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 218.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 273.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 650.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 633.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 290.T + 9.12e5T^{2} \) |
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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
−9.115253547982256477428386323998, −8.013717854581979603216759225150, −7.47802540924517717909209930027, −6.53129976189795132122695275063, −5.83623622731695568509591220780, −4.70815571175172162887322998492, −4.08303854143585233081256027855, −2.70077982420870413419551442276, −2.24389503281110996246387293481, −0.17865093248584971450077545397,
0.17865093248584971450077545397, 2.24389503281110996246387293481, 2.70077982420870413419551442276, 4.08303854143585233081256027855, 4.70815571175172162887322998492, 5.83623622731695568509591220780, 6.53129976189795132122695275063, 7.47802540924517717909209930027, 8.013717854581979603216759225150, 9.115253547982256477428386323998
