| L(s) = 1 | − 2.61·2-s + 1.92·3-s + 4.82·4-s − 2.98·5-s − 5.02·6-s + 0.679·7-s − 7.37·8-s + 0.696·9-s + 7.80·10-s + 2.20·11-s + 9.27·12-s − 2.41·13-s − 1.77·14-s − 5.74·15-s + 9.62·16-s + 2.52·17-s − 1.82·18-s − 1.64·19-s − 14.4·20-s + 1.30·21-s − 5.75·22-s + 0.0392·23-s − 14.1·24-s + 3.92·25-s + 6.32·26-s − 4.42·27-s + 3.27·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 1.11·3-s + 2.41·4-s − 1.33·5-s − 2.05·6-s + 0.256·7-s − 2.60·8-s + 0.232·9-s + 2.46·10-s + 0.663·11-s + 2.67·12-s − 0.671·13-s − 0.474·14-s − 1.48·15-s + 2.40·16-s + 0.613·17-s − 0.429·18-s − 0.376·19-s − 3.22·20-s + 0.285·21-s − 1.22·22-s + 0.00819·23-s − 2.89·24-s + 0.785·25-s + 1.23·26-s − 0.852·27-s + 0.619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 0.679T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 0.0392T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + 3.93T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 + 9.96T + 89T^{2} \) |
| 97 | \( 1 - 0.945T + 97T^{2} \) |
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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
−8.665918744103874487719611218616, −8.147109973024781042394714682409, −7.72661251247012988705293046509, −7.09289913196685983956740414829, −6.08700580660098460798647609520, −4.45468594607642978081127342111, −3.39702861131923152123781488019, −2.59666526004694303948312757547, −1.43417574361808464727299635509, 0,
1.43417574361808464727299635509, 2.59666526004694303948312757547, 3.39702861131923152123781488019, 4.45468594607642978081127342111, 6.08700580660098460798647609520, 7.09289913196685983956740414829, 7.72661251247012988705293046509, 8.147109973024781042394714682409, 8.665918744103874487719611218616
