L(s) = 1 | + 2.10·2-s + 2.41·4-s + 2.10·5-s + 0.870·8-s + 4.41·10-s − 3.84·11-s + 6.82·13-s − 2.99·16-s + 1.23·17-s + 4.41·19-s + 5.07·20-s − 8.07·22-s + 6.81·23-s − 0.585·25-s + 14.3·26-s + 8.40·29-s − 0.656·31-s − 8.04·32-s + 2.58·34-s + 3.24·37-s + 9.27·38-s + 1.82·40-s − 5.07·41-s − 7.07·43-s − 9.27·44-s + 14.3·46-s − 10.6·47-s + ⋯ |
L(s) = 1 | + 1.48·2-s + 1.20·4-s + 0.939·5-s + 0.307·8-s + 1.39·10-s − 1.15·11-s + 1.89·13-s − 0.749·16-s + 0.298·17-s + 1.01·19-s + 1.13·20-s − 1.72·22-s + 1.42·23-s − 0.117·25-s + 2.81·26-s + 1.56·29-s − 0.117·31-s − 1.42·32-s + 0.443·34-s + 0.533·37-s + 1.50·38-s + 0.289·40-s − 0.792·41-s − 1.07·43-s − 1.39·44-s + 2.11·46-s − 1.55·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.374101516\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.374101516\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 0.720T + 83T^{2} \) |
| 89 | \( 1 - 0.149T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−9.751253030865841990199669321060, −8.814224060757540200405115880311, −7.931175386341359093646511434359, −6.68630451777558793754329236746, −6.11252333019577611237154748251, −5.32211948054130676557195257473, −4.75967814185732876595708699491, −3.41103462561610339595769797872, −2.87584121943402312033224766960, −1.46860582479803359462865164825,
1.46860582479803359462865164825, 2.87584121943402312033224766960, 3.41103462561610339595769797872, 4.75967814185732876595708699491, 5.32211948054130676557195257473, 6.11252333019577611237154748251, 6.68630451777558793754329236746, 7.931175386341359093646511434359, 8.814224060757540200405115880311, 9.751253030865841990199669321060