| L(s) = 1 | + 1.30·3-s + 2.30·5-s − 7-s − 1.30·9-s + 6.60·13-s + 3·15-s + 6.60·19-s − 1.30·21-s + 0.302·25-s − 5.60·27-s − 4.30·31-s − 2.30·35-s + 2.60·37-s + 8.60·39-s − 3.90·41-s + 7.30·43-s − 3.00·45-s + 4.60·47-s + 49-s − 3.69·53-s + 8.60·57-s + 9.21·59-s + 7.90·61-s + 1.30·63-s + 15.2·65-s − 1.69·67-s + 7.81·71-s + ⋯ |
| L(s) = 1 | + 0.752·3-s + 1.02·5-s − 0.377·7-s − 0.434·9-s + 1.83·13-s + 0.774·15-s + 1.51·19-s − 0.284·21-s + 0.0605·25-s − 1.07·27-s − 0.772·31-s − 0.389·35-s + 0.428·37-s + 1.37·39-s − 0.610·41-s + 1.11·43-s − 0.447·45-s + 0.671·47-s + 0.142·49-s − 0.507·53-s + 1.13·57-s + 1.19·59-s + 1.01·61-s + 0.164·63-s + 1.88·65-s − 0.207·67-s + 0.927·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.558992343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.558992343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−7.946423738174697943832676474266, −7.16132020828366039380091550104, −6.32860878264655289987567070582, −5.72698731312553703426512554854, −5.33554989996447177568795594665, −3.98368099772383706997406110071, −3.44352690751713961120840084827, −2.70321100912511630441997962618, −1.85726768790780131108194570529, −0.934653797610066712946965104778,
0.934653797610066712946965104778, 1.85726768790780131108194570529, 2.70321100912511630441997962618, 3.44352690751713961120840084827, 3.98368099772383706997406110071, 5.33554989996447177568795594665, 5.72698731312553703426512554854, 6.32860878264655289987567070582, 7.16132020828366039380091550104, 7.946423738174697943832676474266
