| L(s) = 1 | − 1.31·3-s + 5-s + 3.52·7-s − 1.28·9-s − 1.52·11-s − 6.02·13-s − 1.31·15-s + 5.05·17-s − 19-s − 4.62·21-s + 2.28·23-s + 25-s + 5.61·27-s − 4.62·29-s − 7.80·31-s + 1.99·33-s + 3.52·35-s + 9.39·37-s + 7.89·39-s + 2.70·43-s − 1.28·45-s − 5.33·47-s + 5.42·49-s − 6.62·51-s + 9.82·53-s − 1.52·55-s + 1.31·57-s + ⋯ |
| L(s) = 1 | − 0.756·3-s + 0.447·5-s + 1.33·7-s − 0.426·9-s − 0.459·11-s − 1.66·13-s − 0.338·15-s + 1.22·17-s − 0.229·19-s − 1.00·21-s + 0.475·23-s + 0.200·25-s + 1.08·27-s − 0.858·29-s − 1.40·31-s + 0.348·33-s + 0.595·35-s + 1.54·37-s + 1.26·39-s + 0.413·43-s − 0.190·45-s − 0.777·47-s + 0.775·49-s − 0.927·51-s + 1.34·53-s − 0.205·55-s + 0.173·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 + 6.02T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 9.82T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 + 0.193T + 71T^{2} \) |
| 73 | \( 1 + 9.47T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−7.62462007995183278701890524063, −7.18938713130019500578491954424, −6.03416018038860541746617024805, −5.47022665642890416962305120704, −5.03607957678059310821701853048, −4.36503495977556104260839236452, −3.05036412249902145274192222559, −2.26054500243506590592634442512, −1.29817839729038900233098437285, 0,
1.29817839729038900233098437285, 2.26054500243506590592634442512, 3.05036412249902145274192222559, 4.36503495977556104260839236452, 5.03607957678059310821701853048, 5.47022665642890416962305120704, 6.03416018038860541746617024805, 7.18938713130019500578491954424, 7.62462007995183278701890524063
