| L(s) = 1 | − 1.69·2-s + 0.888·4-s − 3.58·5-s + 1.88·8-s + 6.09·10-s + 2.81·11-s − 13-s − 4.98·16-s − 4.11·17-s + 0.888·19-s − 3.18·20-s − 4.77·22-s − 5.87·23-s + 7.87·25-s + 1.69·26-s + 1.69·29-s + 6.98·31-s + 4.69·32-s + 6.98·34-s + 4.76·37-s − 1.51·38-s − 6.77·40-s − 5.41·41-s + 5.21·43-s + 2.49·44-s + 9.98·46-s − 2.66·47-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.444·4-s − 1.60·5-s + 0.667·8-s + 1.92·10-s + 0.847·11-s − 0.277·13-s − 1.24·16-s − 0.997·17-s + 0.203·19-s − 0.713·20-s − 1.01·22-s − 1.22·23-s + 1.57·25-s + 0.333·26-s + 0.315·29-s + 1.25·31-s + 0.830·32-s + 1.19·34-s + 0.783·37-s − 0.245·38-s − 1.07·40-s − 0.845·41-s + 0.794·43-s + 0.376·44-s + 1.47·46-s − 0.388·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 0.888T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 0.123T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 7.32T + 97T^{2} \) |
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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
−8.202345760456439430165990313379, −7.61727622987236709503409767785, −6.92441100112976578321844199507, −6.23531678131242638370973943648, −4.73295429878113817897898775001, −4.31120456375264468763613519346, −3.49582154834270131951969696871, −2.23945444273805631871479100518, −0.984264107350135747147727588341, 0,
0.984264107350135747147727588341, 2.23945444273805631871479100518, 3.49582154834270131951969696871, 4.31120456375264468763613519346, 4.73295429878113817897898775001, 6.23531678131242638370973943648, 6.92441100112976578321844199507, 7.61727622987236709503409767785, 8.202345760456439430165990313379
