| L(s) = 1 | − 0.916·2-s − 0.984·3-s − 1.15·4-s + 5-s + 0.902·6-s − 4.70·7-s + 2.89·8-s − 2.03·9-s − 0.916·10-s + 6.16·11-s + 1.14·12-s − 0.271·13-s + 4.30·14-s − 0.984·15-s − 0.334·16-s − 4.97·17-s + 1.86·18-s − 3.14·19-s − 1.15·20-s + 4.63·21-s − 5.64·22-s + 5.36·23-s − 2.85·24-s + 25-s + 0.248·26-s + 4.95·27-s + 5.45·28-s + ⋯ |
| L(s) = 1 | − 0.648·2-s − 0.568·3-s − 0.579·4-s + 0.447·5-s + 0.368·6-s − 1.77·7-s + 1.02·8-s − 0.676·9-s − 0.289·10-s + 1.85·11-s + 0.329·12-s − 0.0753·13-s + 1.15·14-s − 0.254·15-s − 0.0837·16-s − 1.20·17-s + 0.438·18-s − 0.721·19-s − 0.259·20-s + 1.01·21-s − 1.20·22-s + 1.11·23-s − 0.582·24-s + 0.200·25-s + 0.0488·26-s + 0.953·27-s + 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.916T + 2T^{2} \) |
| 3 | \( 1 + 0.984T + 3T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 + 0.271T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 3.14T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 0.802T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 7.49T + 53T^{2} \) |
| 59 | \( 1 + 0.0479T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−8.387038492631470896755688444090, −6.91685026126426154644718782164, −6.72833698267458773520430667632, −6.01447943798702991520379390034, −5.16308441180794722747405307894, −4.14237410165003958465925804042, −3.51289125750523096720494189166, −2.33872865664446497317407008106, −0.989604762083926878963878330623, 0,
0.989604762083926878963878330623, 2.33872865664446497317407008106, 3.51289125750523096720494189166, 4.14237410165003958465925804042, 5.16308441180794722747405307894, 6.01447943798702991520379390034, 6.72833698267458773520430667632, 6.91685026126426154644718782164, 8.387038492631470896755688444090
