L(s) = 1 | + 1.37·2-s − 6.11·4-s − 10.3·5-s − 5.11·7-s − 19.3·8-s − 14.2·10-s − 55.9·11-s + 37.5·13-s − 7.02·14-s + 22.3·16-s − 23.6·17-s + 39.0·19-s + 63.4·20-s − 76.8·22-s − 71.0·23-s − 17.4·25-s + 51.5·26-s + 31.2·28-s + 28.3·29-s + 12.8·31-s + 185.·32-s − 32.4·34-s + 53.0·35-s − 180.·37-s + 53.5·38-s + 200.·40-s + 215.·41-s + ⋯ |
L(s) = 1 | + 0.485·2-s − 0.764·4-s − 0.927·5-s − 0.276·7-s − 0.856·8-s − 0.450·10-s − 1.53·11-s + 0.801·13-s − 0.134·14-s + 0.349·16-s − 0.337·17-s + 0.471·19-s + 0.709·20-s − 0.744·22-s − 0.644·23-s − 0.139·25-s + 0.389·26-s + 0.211·28-s + 0.181·29-s + 0.0746·31-s + 1.02·32-s − 0.163·34-s + 0.256·35-s − 0.800·37-s + 0.228·38-s + 0.794·40-s + 0.820·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 - 1.37T + 8T^{2} \) |
| 5 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 + 5.11T + 343T^{2} \) |
| 11 | \( 1 + 55.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 71.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 28.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 61.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 61.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 270.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 713.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 404.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 75.0T + 9.12e5T^{2} \) |
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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
−13.27821050672043648651569477118, −12.48645682542639537645263523440, −11.27989395502828336414464370008, −10.01524481259845226850486140110, −8.621861034883653846468964614760, −7.67062842802970788318753774702, −5.86024017939662218629439332761, −4.53682533596965237153114113345, −3.25309316451567943879791015046, 0,
3.25309316451567943879791015046, 4.53682533596965237153114113345, 5.86024017939662218629439332761, 7.67062842802970788318753774702, 8.621861034883653846468964614760, 10.01524481259845226850486140110, 11.27989395502828336414464370008, 12.48645682542639537645263523440, 13.27821050672043648651569477118