| L(s) = 1 | + 2.23·3-s + 2.28·5-s − 3.70·7-s + 2.00·9-s − 0.540·11-s − 6.23·13-s + 5.11·15-s + 3.70·17-s − 0.333·19-s − 8.27·21-s + 0.236·25-s − 2.23·27-s − 8.70·29-s − 7.94·31-s − 1.20·33-s − 8.47·35-s − 9.69·37-s − 13.9·39-s + 2.70·41-s + 8.48·43-s + 4.57·45-s − 9.47·47-s + 6.70·49-s + 8.27·51-s − 6.65·53-s − 1.23·55-s − 0.746·57-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 1.02·5-s − 1.39·7-s + 0.666·9-s − 0.162·11-s − 1.72·13-s + 1.32·15-s + 0.897·17-s − 0.0765·19-s − 1.80·21-s + 0.0472·25-s − 0.430·27-s − 1.61·29-s − 1.42·31-s − 0.210·33-s − 1.43·35-s − 1.59·37-s − 2.23·39-s + 0.422·41-s + 1.29·43-s + 0.682·45-s − 1.38·47-s + 0.958·49-s + 1.15·51-s − 0.914·53-s − 0.166·55-s − 0.0988·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 + 0.540T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 0.333T + 19T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 9.69T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 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
−7.996509450847638803701776252957, −7.34014190010723759683779201770, −6.76599495845184453499681834722, −5.68677194957398580909361277785, −5.27958145499113637878211808351, −3.86269385454848696863977099769, −3.27080669612796866221863868753, −2.45646822273524629040634028063, −1.88133503390189803246768829430, 0,
1.88133503390189803246768829430, 2.45646822273524629040634028063, 3.27080669612796866221863868753, 3.86269385454848696863977099769, 5.27958145499113637878211808351, 5.68677194957398580909361277785, 6.76599495845184453499681834722, 7.34014190010723759683779201770, 7.996509450847638803701776252957
