| L(s) = 1 | + 2.37·2-s + 3.62·4-s + 1.22·5-s + 3.85·8-s + 2.91·10-s − 2.66·11-s + 3.62·13-s + 1.88·16-s + 6.73·17-s − 2.51·19-s + 4.45·20-s − 6.31·22-s + 7.99·23-s − 3.49·25-s + 8.59·26-s − 2.24·29-s + 10.2·31-s − 3.22·32-s + 15.9·34-s − 3.53·37-s − 5.96·38-s + 4.73·40-s + 1.86·41-s + 5.13·43-s − 9.64·44-s + 18.9·46-s − 2.14·47-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.81·4-s + 0.549·5-s + 1.36·8-s + 0.921·10-s − 0.802·11-s + 1.00·13-s + 0.472·16-s + 1.63·17-s − 0.576·19-s + 0.995·20-s − 1.34·22-s + 1.66·23-s − 0.698·25-s + 1.68·26-s − 0.417·29-s + 1.83·31-s − 0.570·32-s + 2.73·34-s − 0.581·37-s − 0.967·38-s + 0.748·40-s + 0.291·41-s + 0.783·43-s − 1.45·44-s + 2.79·46-s − 0.313·47-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
| \(L(1)\) |
\(\approx\) |
\(6.199385678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.199385678\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 - 5.13T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 + 2.97T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 + 0.313T + 79T^{2} \) |
| 83 | \( 1 - 7.69T + 83T^{2} \) |
| 89 | \( 1 + 7.19T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.264016938824481290228083346029, −7.52682632577267670645690314061, −6.61871455542520632773611231573, −6.01055280350625975904708096597, −5.40490235577959264150626994386, −4.82041202547137967204795739007, −3.84622118287558747075970123859, −3.13718450495830824514778001766, −2.40813954636209964861114108291, −1.21790938558758227191926983072,
1.21790938558758227191926983072, 2.40813954636209964861114108291, 3.13718450495830824514778001766, 3.84622118287558747075970123859, 4.82041202547137967204795739007, 5.40490235577959264150626994386, 6.01055280350625975904708096597, 6.61871455542520632773611231573, 7.52682632577267670645690314061, 8.264016938824481290228083346029
