| L(s) = 1 | + 13.1·5-s + 15.6·7-s + 55.7·11-s + 13·13-s − 23.4·17-s − 25.6·19-s + 189.·23-s + 47.2·25-s + 236.·29-s + 47.0·31-s + 204.·35-s − 154.·37-s + 34.6·41-s − 398.·43-s − 582.·47-s − 99.1·49-s + 361.·53-s + 731.·55-s − 396.·59-s − 211.·61-s + 170.·65-s + 85.8·67-s + 651.·71-s + 927.·73-s + 870.·77-s − 1.11e3·79-s − 391.·83-s + ⋯ |
| L(s) = 1 | + 1.17·5-s + 0.843·7-s + 1.52·11-s + 0.277·13-s − 0.334·17-s − 0.309·19-s + 1.71·23-s + 0.377·25-s + 1.51·29-s + 0.272·31-s + 0.989·35-s − 0.687·37-s + 0.132·41-s − 1.41·43-s − 1.80·47-s − 0.289·49-s + 0.935·53-s + 1.79·55-s − 0.875·59-s − 0.443·61-s + 0.325·65-s + 0.156·67-s + 1.08·71-s + 1.48·73-s + 1.28·77-s − 1.59·79-s − 0.517·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.464099833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.464099833\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 13.1T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 23.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 396.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 651.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 927.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 391.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 745.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 173.T + 9.12e5T^{2} \) |
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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
−9.600076179397394488234730721532, −8.895911072595924749869130179301, −8.220984110160874276207255023313, −6.78834215771319457470197041577, −6.43181432700920144276756550578, −5.24698985792822060531888437767, −4.51232406615639476375093926393, −3.20842440946067003174356554093, −1.88325420162029268508603276097, −1.13746870841726514792562803097,
1.13746870841726514792562803097, 1.88325420162029268508603276097, 3.20842440946067003174356554093, 4.51232406615639476375093926393, 5.24698985792822060531888437767, 6.43181432700920144276756550578, 6.78834215771319457470197041577, 8.220984110160874276207255023313, 8.895911072595924749869130179301, 9.600076179397394488234730721532
