L(s) = 1 | − 2.45·2-s + 4.04·4-s + 3.33·5-s − 3.69·7-s − 5.03·8-s − 8.20·10-s − 0.270·11-s + 9.08·14-s + 4.29·16-s + 4.04·17-s − 7.15·19-s + 13.5·20-s + 0.664·22-s + 2.79·23-s + 6.13·25-s − 14.9·28-s − 7.83·29-s − 2.14·31-s − 0.487·32-s − 9.93·34-s − 12.3·35-s − 4.37·37-s + 17.6·38-s − 16.8·40-s − 7.61·41-s − 3.89·43-s − 1.09·44-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.02·4-s + 1.49·5-s − 1.39·7-s − 1.78·8-s − 2.59·10-s − 0.0815·11-s + 2.42·14-s + 1.07·16-s + 0.980·17-s − 1.64·19-s + 3.02·20-s + 0.141·22-s + 0.583·23-s + 1.22·25-s − 2.82·28-s − 1.45·29-s − 0.385·31-s − 0.0861·32-s − 1.70·34-s − 2.08·35-s − 0.719·37-s + 2.85·38-s − 2.65·40-s − 1.18·41-s − 0.593·43-s − 0.165·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 0.270T + 11T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 23 | \( 1 - 2.79T + 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 - 0.877T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 - 3.40T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 + 7.59T + 71T^{2} \) |
| 73 | \( 1 - 0.405T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.167770772784264880493039349812, −8.646802485176583542872214698596, −7.54406111260861141897583092343, −6.65147435542939920414151934418, −6.24417363144239276125351478263, −5.31194935683536856353881318696, −3.47400178013755761644798392882, −2.40827694742778115771991720295, −1.55671476947290960686366092336, 0,
1.55671476947290960686366092336, 2.40827694742778115771991720295, 3.47400178013755761644798392882, 5.31194935683536856353881318696, 6.24417363144239276125351478263, 6.65147435542939920414151934418, 7.54406111260861141897583092343, 8.646802485176583542872214698596, 9.167770772784264880493039349812