| L(s) = 1 | − 5.60·2-s − 7.98·3-s + 23.4·4-s − 5.27·5-s + 44.7·6-s + 7·7-s − 86.7·8-s + 36.7·9-s + 29.5·10-s − 50.0·11-s − 187.·12-s − 60.3·13-s − 39.2·14-s + 42.0·15-s + 298.·16-s − 87.5·17-s − 206.·18-s − 23.8·19-s − 123.·20-s − 55.8·21-s + 281.·22-s + 23·23-s + 692.·24-s − 97.2·25-s + 338.·26-s − 77.8·27-s + 164.·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 1.53·3-s + 2.93·4-s − 0.471·5-s + 3.04·6-s + 0.377·7-s − 3.83·8-s + 1.36·9-s + 0.934·10-s − 1.37·11-s − 4.50·12-s − 1.28·13-s − 0.749·14-s + 0.724·15-s + 4.67·16-s − 1.24·17-s − 2.69·18-s − 0.287·19-s − 1.38·20-s − 0.580·21-s + 2.72·22-s + 0.208·23-s + 5.89·24-s − 0.777·25-s + 2.55·26-s − 0.554·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1305431716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1305431716\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 5.60T + 8T^{2} \) |
| 3 | \( 1 + 7.98T + 27T^{2} \) |
| 5 | \( 1 + 5.27T + 125T^{2} \) |
| 11 | \( 1 + 50.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 102.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 43.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 165.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 606.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 358.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 699.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 255.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 43.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 896.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 56.3T + 9.12e5T^{2} \) |
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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
−11.84127664173250162531695891730, −11.08834025310922745318875879325, −10.52341721369445726007278311689, −9.560670816209608978735223614493, −8.167689947374689944652251870027, −7.36008133627449215169925413512, −6.37617335821426044812059421537, −5.08659949497351894994538905413, −2.29320278579173792343394336858, −0.38935439294581919508885722813,
0.38935439294581919508885722813, 2.29320278579173792343394336858, 5.08659949497351894994538905413, 6.37617335821426044812059421537, 7.36008133627449215169925413512, 8.167689947374689944652251870027, 9.560670816209608978735223614493, 10.52341721369445726007278311689, 11.08834025310922745318875879325, 11.84127664173250162531695891730
