L(s) = 1 | + 0.904·3-s + 3.10·5-s + 1.70·7-s − 2.18·9-s − 6.05·11-s + 1.97·13-s + 2.80·15-s + 0.802·17-s + 2.24·19-s + 1.54·21-s − 2.79·23-s + 4.63·25-s − 4.68·27-s + 9.23·29-s + 7.61·31-s − 5.48·33-s + 5.30·35-s + 3.25·37-s + 1.78·39-s + 3.62·41-s − 0.635·43-s − 6.76·45-s + 47-s − 4.08·49-s + 0.725·51-s − 4.24·53-s − 18.8·55-s + ⋯ |
L(s) = 1 | + 0.522·3-s + 1.38·5-s + 0.645·7-s − 0.727·9-s − 1.82·11-s + 0.547·13-s + 0.725·15-s + 0.194·17-s + 0.515·19-s + 0.337·21-s − 0.581·23-s + 0.926·25-s − 0.902·27-s + 1.71·29-s + 1.36·31-s − 0.954·33-s + 0.896·35-s + 0.535·37-s + 0.286·39-s + 0.566·41-s − 0.0969·43-s − 1.00·45-s + 0.145·47-s − 0.583·49-s + 0.101·51-s − 0.583·53-s − 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.126290033\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.126290033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 3 | \( 1 - 0.904T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 0.802T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 3.62T + 41T^{2} \) |
| 43 | \( 1 + 0.635T + 43T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 0.818T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.193264235851538221109555700600, −7.64743568347244483574892750175, −6.44914333295194108108501823905, −5.92988090833364857783025397080, −5.21186914448418107208437045050, −4.72073425236610297579065666468, −3.36304126778614152900032357241, −2.57137436320194031732318087171, −2.15337934920042101677154663481, −0.909708294562438124989072847372,
0.909708294562438124989072847372, 2.15337934920042101677154663481, 2.57137436320194031732318087171, 3.36304126778614152900032357241, 4.72073425236610297579065666468, 5.21186914448418107208437045050, 5.92988090833364857783025397080, 6.44914333295194108108501823905, 7.64743568347244483574892750175, 8.193264235851538221109555700600