| L(s) = 1 | + 0.732·3-s − 5-s + 2·7-s − 2.46·9-s − 3.46·11-s − 0.732·13-s − 0.732·15-s + 3.46·17-s − 19-s + 1.46·21-s + 3.46·23-s + 25-s − 4·27-s + 3.46·29-s + 5.46·31-s − 2.53·33-s − 2·35-s − 3.26·37-s − 0.535·39-s − 6·41-s − 8.92·43-s + 2.46·45-s − 0.928·47-s − 3·49-s + 2.53·51-s + 7.26·53-s + 3.46·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s + 0.755·7-s − 0.821·9-s − 1.04·11-s − 0.203·13-s − 0.189·15-s + 0.840·17-s − 0.229·19-s + 0.319·21-s + 0.722·23-s + 0.200·25-s − 0.769·27-s + 0.643·29-s + 0.981·31-s − 0.441·33-s − 0.338·35-s − 0.537·37-s − 0.0858·39-s − 0.937·41-s − 1.36·43-s + 0.367·45-s − 0.135·47-s − 0.428·49-s + 0.355·51-s + 0.998·53-s + 0.467·55-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 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.84501182151761096713105601714, −7.20883961411822483944228841817, −6.33379901165874026266606079897, −5.24869759704150111834635173248, −5.06957600138719121341195212249, −3.97894299381630711573307858105, −3.06566300237167678146767588062, −2.53590389288630047714337459904, −1.35235105511538362174068784132, 0,
1.35235105511538362174068784132, 2.53590389288630047714337459904, 3.06566300237167678146767588062, 3.97894299381630711573307858105, 5.06957600138719121341195212249, 5.24869759704150111834635173248, 6.33379901165874026266606079897, 7.20883961411822483944228841817, 7.84501182151761096713105601714
