| L(s) = 1 | − 1.10·2-s + 3-s − 0.772·4-s + 2.82·5-s − 1.10·6-s + 3.07·8-s + 9-s − 3.12·10-s + 5.50·11-s − 0.772·12-s + 13-s + 2.82·15-s − 1.85·16-s + 5.68·17-s − 1.10·18-s + 2.85·19-s − 2.17·20-s − 6.10·22-s − 4.90·23-s + 3.07·24-s + 2.97·25-s − 1.10·26-s + 27-s + 7.57·29-s − 3.12·30-s − 6.33·31-s − 4.08·32-s + ⋯ |
| L(s) = 1 | − 0.783·2-s + 0.577·3-s − 0.386·4-s + 1.26·5-s − 0.452·6-s + 1.08·8-s + 0.333·9-s − 0.989·10-s + 1.66·11-s − 0.222·12-s + 0.277·13-s + 0.728·15-s − 0.464·16-s + 1.37·17-s − 0.261·18-s + 0.656·19-s − 0.487·20-s − 1.30·22-s − 1.02·23-s + 0.627·24-s + 0.594·25-s − 0.217·26-s + 0.192·27-s + 1.40·29-s − 0.571·30-s − 1.13·31-s − 0.721·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922389849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922389849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 + 9.90T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 2.43T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 6.68T + 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 6.13T + 97T^{2} \) |
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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
−9.262875773183343462535543055501, −8.650278761050593047240849579505, −7.87376708785388479651814751964, −6.97478392191799931596450686858, −6.09760650939878196660341490027, −5.26105842826723202578343688175, −4.14645332058451061249425017480, −3.28469524352250821264678836790, −1.80191683249567918048936962061, −1.19624813318876455203423792625,
1.19624813318876455203423792625, 1.80191683249567918048936962061, 3.28469524352250821264678836790, 4.14645332058451061249425017480, 5.26105842826723202578343688175, 6.09760650939878196660341490027, 6.97478392191799931596450686858, 7.87376708785388479651814751964, 8.650278761050593047240849579505, 9.262875773183343462535543055501
