| L(s) = 1 | + 0.639·2-s − 3-s − 1.59·4-s − 2.86·5-s − 0.639·6-s − 7-s − 2.29·8-s + 9-s − 1.83·10-s − 2.14·11-s + 1.59·12-s − 4.28·13-s − 0.639·14-s + 2.86·15-s + 1.71·16-s − 2.78·17-s + 0.639·18-s − 4.83·19-s + 4.55·20-s + 21-s − 1.37·22-s − 0.469·23-s + 2.29·24-s + 3.20·25-s − 2.74·26-s − 27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | + 0.452·2-s − 0.577·3-s − 0.795·4-s − 1.28·5-s − 0.261·6-s − 0.377·7-s − 0.812·8-s + 0.333·9-s − 0.579·10-s − 0.646·11-s + 0.459·12-s − 1.18·13-s − 0.170·14-s + 0.739·15-s + 0.427·16-s − 0.675·17-s + 0.150·18-s − 1.10·19-s + 1.01·20-s + 0.218·21-s − 0.292·22-s − 0.0978·23-s + 0.468·24-s + 0.640·25-s − 0.537·26-s − 0.192·27-s + 0.300·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07402909947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07402909947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.639T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 + 0.469T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 + 4.49T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 + 0.223T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 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
−8.735710912596294892477589625246, −8.117824555125039836169577990254, −7.24406347613930646416445231451, −6.62085962492749851077179816094, −5.40179245551864474124897139152, −5.00396014848180515661249744452, −4.03057576396879588468116689818, −3.58408723039858302558455627307, −2.25665990932349670634931314037, −0.15322825595356315755876329603,
0.15322825595356315755876329603, 2.25665990932349670634931314037, 3.58408723039858302558455627307, 4.03057576396879588468116689818, 5.00396014848180515661249744452, 5.40179245551864474124897139152, 6.62085962492749851077179816094, 7.24406347613930646416445231451, 8.117824555125039836169577990254, 8.735710912596294892477589625246
