L(s) = 1 | − 2-s − 2.19·3-s + 4-s − 5-s + 2.19·6-s − 4.31·7-s − 8-s + 1.81·9-s + 10-s − 0.0662·11-s − 2.19·12-s + 0.0541·13-s + 4.31·14-s + 2.19·15-s + 16-s − 4.82·17-s − 1.81·18-s + 3.98·19-s − 20-s + 9.46·21-s + 0.0662·22-s − 5.13·23-s + 2.19·24-s + 25-s − 0.0541·26-s + 2.60·27-s − 4.31·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.5·4-s − 0.447·5-s + 0.895·6-s − 1.63·7-s − 0.353·8-s + 0.604·9-s + 0.316·10-s − 0.0199·11-s − 0.633·12-s + 0.0150·13-s + 1.15·14-s + 0.566·15-s + 0.250·16-s − 1.16·17-s − 0.427·18-s + 0.913·19-s − 0.223·20-s + 2.06·21-s + 0.0141·22-s − 1.07·23-s + 0.447·24-s + 0.200·25-s − 0.0106·26-s + 0.500·27-s − 0.815·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 0.0662T + 11T^{2} \) |
| 13 | \( 1 - 0.0541T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 5.13T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 + 0.606T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + 1.86T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 2.77T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.57636636956719421948359474098, −6.83464903451634859952470930787, −6.45722586253611489651093863572, −5.80569193003586667457522227831, −5.03323460644911339544987091609, −3.98135992178790934983073599309, −3.23333225574615638578291705808, −2.21935221589436883656586239561, −0.75221543973150598387334305872, 0,
0.75221543973150598387334305872, 2.21935221589436883656586239561, 3.23333225574615638578291705808, 3.98135992178790934983073599309, 5.03323460644911339544987091609, 5.80569193003586667457522227831, 6.45722586253611489651093863572, 6.83464903451634859952470930787, 7.57636636956719421948359474098