| L(s) = 1 | − 1.24·2-s − 2.63·3-s − 0.439·4-s − 5-s + 3.29·6-s + 1.69·7-s + 3.04·8-s + 3.93·9-s + 1.24·10-s + 11-s + 1.15·12-s + 1.36·13-s − 2.11·14-s + 2.63·15-s − 2.92·16-s − 3.21·17-s − 4.92·18-s + 0.489·19-s + 0.439·20-s − 4.45·21-s − 1.24·22-s − 2.30·23-s − 8.02·24-s + 25-s − 1.70·26-s − 2.47·27-s − 0.743·28-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 1.52·3-s − 0.219·4-s − 0.447·5-s + 1.34·6-s + 0.639·7-s + 1.07·8-s + 1.31·9-s + 0.395·10-s + 0.301·11-s + 0.334·12-s + 0.377·13-s − 0.565·14-s + 0.680·15-s − 0.732·16-s − 0.779·17-s − 1.15·18-s + 0.112·19-s + 0.0982·20-s − 0.972·21-s − 0.266·22-s − 0.480·23-s − 1.63·24-s + 0.200·25-s − 0.333·26-s − 0.476·27-s − 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 2.63T + 3T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 - 0.489T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 + 5.93T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 8.04T + 97T^{2} \) |
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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
−8.169662326521056221147059339131, −7.38477234344311614936592479100, −6.67470551909026339928712945194, −5.90731853511795648396096967582, −5.00845187911743151806881690558, −4.52845869152748500257888620472, −3.69269331988478866401941516515, −1.93852707020579227805848095875, −0.976839863069535120990962814022, 0,
0.976839863069535120990962814022, 1.93852707020579227805848095875, 3.69269331988478866401941516515, 4.52845869152748500257888620472, 5.00845187911743151806881690558, 5.90731853511795648396096967582, 6.67470551909026339928712945194, 7.38477234344311614936592479100, 8.169662326521056221147059339131
