| L(s) = 1 | − 1.85·3-s + 2.88·5-s + 1.11·7-s + 0.447·9-s − 4.87·11-s + 3.77·13-s − 5.36·15-s − 0.0587·17-s − 2.60·19-s − 2.06·21-s − 2.33·23-s + 3.34·25-s + 4.73·27-s + 6.17·29-s − 4.36·31-s + 9.05·33-s + 3.21·35-s − 4.47·37-s − 7.01·39-s − 2.97·41-s − 10.9·43-s + 1.29·45-s + 8.04·47-s − 5.76·49-s + 0.109·51-s − 11.7·53-s − 14.0·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 1.29·5-s + 0.420·7-s + 0.149·9-s − 1.47·11-s + 1.04·13-s − 1.38·15-s − 0.0142·17-s − 0.596·19-s − 0.450·21-s − 0.486·23-s + 0.669·25-s + 0.912·27-s + 1.14·29-s − 0.784·31-s + 1.57·33-s + 0.543·35-s − 0.735·37-s − 1.12·39-s − 0.464·41-s − 1.66·43-s + 0.192·45-s + 1.17·47-s − 0.823·49-s + 0.0152·51-s − 1.60·53-s − 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.85T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.0587T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 8.04T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.31T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 0.407T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 4.02T + 97T^{2} \) |
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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
−8.303294150079734566643014405139, −7.15521907303626781481647845577, −6.28568831275609874093814749642, −5.89454258007954433510767633588, −5.19967940761985156215402637401, −4.68765589893273956315003958533, −3.31935384189773238359405938122, −2.29120072874412688130110059920, −1.41911589043574703010848201208, 0,
1.41911589043574703010848201208, 2.29120072874412688130110059920, 3.31935384189773238359405938122, 4.68765589893273956315003958533, 5.19967940761985156215402637401, 5.89454258007954433510767633588, 6.28568831275609874093814749642, 7.15521907303626781481647845577, 8.303294150079734566643014405139
