| L(s) = 1 | − 1.84·2-s + 1.39·4-s + 1.33·5-s + 1.12·8-s − 2.45·10-s + 1.51·11-s + 5.17·13-s − 4.84·16-s − 1.54·17-s + 2.50·19-s + 1.85·20-s − 2.78·22-s − 7.36·23-s − 3.21·25-s − 9.53·26-s + 0.0619·29-s − 3.84·31-s + 6.68·32-s + 2.85·34-s + 0.563·37-s − 4.61·38-s + 1.49·40-s − 9.02·41-s − 10.1·43-s + 2.10·44-s + 13.5·46-s − 9.51·47-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.695·4-s + 0.596·5-s + 0.396·8-s − 0.777·10-s + 0.456·11-s + 1.43·13-s − 1.21·16-s − 0.375·17-s + 0.574·19-s + 0.414·20-s − 0.593·22-s − 1.53·23-s − 0.643·25-s − 1.86·26-s + 0.0115·29-s − 0.691·31-s + 1.18·32-s + 0.489·34-s + 0.0925·37-s − 0.747·38-s + 0.236·40-s − 1.40·41-s − 1.55·43-s + 0.317·44-s + 1.99·46-s − 1.38·47-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 | \( 1 + 1.84T + 2T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 + 7.36T + 23T^{2} \) |
| 29 | \( 1 - 0.0619T + 29T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 - 0.563T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.319341015529661680349183899737, −7.61110842696076561000414849915, −6.65473093782183329368806115746, −6.16114108664223129213688322741, −5.22926004508047472195378940373, −4.17220298027898560580371449438, −3.34562202753758605914047295934, −1.89810784562013162303119005251, −1.44752282223923598176683375783, 0,
1.44752282223923598176683375783, 1.89810784562013162303119005251, 3.34562202753758605914047295934, 4.17220298027898560580371449438, 5.22926004508047472195378940373, 6.16114108664223129213688322741, 6.65473093782183329368806115746, 7.61110842696076561000414849915, 8.319341015529661680349183899737
