| L(s) = 1 | − 1.51·2-s − 3.07·3-s + 0.295·4-s + 4.66·6-s − 2.65·7-s + 2.58·8-s + 6.48·9-s − 1.07·11-s − 0.910·12-s − 4.95·13-s + 4.02·14-s − 4.50·16-s + 6.67·17-s − 9.82·18-s − 3.61·19-s + 8.19·21-s + 1.63·22-s − 8.64·23-s − 7.95·24-s + 7.50·26-s − 10.7·27-s − 0.786·28-s − 7.52·29-s − 5.89·31-s + 1.65·32-s + 3.31·33-s − 10.1·34-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 1.77·3-s + 0.147·4-s + 1.90·6-s − 1.00·7-s + 0.913·8-s + 2.16·9-s − 0.324·11-s − 0.262·12-s − 1.37·13-s + 1.07·14-s − 1.12·16-s + 1.61·17-s − 2.31·18-s − 0.829·19-s + 1.78·21-s + 0.347·22-s − 1.80·23-s − 1.62·24-s + 1.47·26-s − 2.06·27-s − 0.148·28-s − 1.39·29-s − 1.05·31-s + 0.293·32-s + 0.577·33-s − 1.73·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1138315774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1138315774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 8.64T + 23T^{2} \) |
| 29 | \( 1 + 7.52T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 2.19T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.36T + 61T^{2} \) |
| 67 | \( 1 - 2.13T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 9.96T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 0.239T + 89T^{2} \) |
| 97 | \( 1 + 4.63T + 97T^{2} \) |
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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
−9.831268353243435812307054837658, −9.434751031401972288781188644519, −7.910831902480940818560161203065, −7.39971144577797055410102507530, −6.45115104660503564001737635714, −5.65131831423437677336948680978, −4.87184144809231838391835609572, −3.77112008729838304526462795649, −1.86978977320293701931951983128, −0.31579852545871710377089778076,
0.31579852545871710377089778076, 1.86978977320293701931951983128, 3.77112008729838304526462795649, 4.87184144809231838391835609572, 5.65131831423437677336948680978, 6.45115104660503564001737635714, 7.39971144577797055410102507530, 7.910831902480940818560161203065, 9.434751031401972288781188644519, 9.831268353243435812307054837658
