L(s) = 1 | + 0.254·2-s − 1.93·4-s − 3.68·5-s − 8-s − 0.935·10-s − 11-s − 2.18·13-s + 3.61·16-s + 3.36·17-s + 3.68·19-s + 7.12·20-s − 0.254·22-s + 8.55·25-s − 0.556·26-s − 10.0·29-s + 8.37·31-s + 2.91·32-s + 0.854·34-s + 0.189·37-s + 0.935·38-s + 3.68·40-s + 8.37·41-s − 8.37·43-s + 1.93·44-s − 5.17·47-s + 2.17·50-s + 4.23·52-s + ⋯ |
L(s) = 1 | + 0.179·2-s − 0.967·4-s − 1.64·5-s − 0.353·8-s − 0.295·10-s − 0.301·11-s − 0.607·13-s + 0.904·16-s + 0.815·17-s + 0.844·19-s + 1.59·20-s − 0.0541·22-s + 1.71·25-s − 0.109·26-s − 1.86·29-s + 1.50·31-s + 0.516·32-s + 0.146·34-s + 0.0311·37-s + 0.151·38-s + 0.582·40-s + 1.30·41-s − 1.27·43-s + 0.291·44-s − 0.754·47-s + 0.307·50-s + 0.587·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.254T + 2T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 8.37T + 31T^{2} \) |
| 37 | \( 1 - 0.189T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 1.36T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 + 0.379T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 2.98T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 0.637T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−7.969947436538252114514921378609, −7.46448275703144170817256386728, −6.55729537911752973057139520594, −5.32406923158942771486042257784, −5.02430778108109053936225814814, −3.98010934941975906057188754863, −3.63921215250076698459508741120, −2.71249270657689861095627765566, −0.993569431080683108367887104559, 0,
0.993569431080683108367887104559, 2.71249270657689861095627765566, 3.63921215250076698459508741120, 3.98010934941975906057188754863, 5.02430778108109053936225814814, 5.32406923158942771486042257784, 6.55729537911752973057139520594, 7.46448275703144170817256386728, 7.969947436538252114514921378609