| L(s) = 1 | + 1.85·2-s + 2.30·3-s + 1.44·4-s + 5-s + 4.27·6-s − 7-s − 1.03·8-s + 2.30·9-s + 1.85·10-s − 4.66·11-s + 3.31·12-s − 5.46·13-s − 1.85·14-s + 2.30·15-s − 4.80·16-s + 3.19·17-s + 4.27·18-s + 0.760·19-s + 1.44·20-s − 2.30·21-s − 8.66·22-s + 3.09·23-s − 2.38·24-s + 25-s − 10.1·26-s − 1.60·27-s − 1.44·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 1.32·3-s + 0.720·4-s + 0.447·5-s + 1.74·6-s − 0.377·7-s − 0.366·8-s + 0.767·9-s + 0.586·10-s − 1.40·11-s + 0.957·12-s − 1.51·13-s − 0.495·14-s + 0.594·15-s − 1.20·16-s + 0.774·17-s + 1.00·18-s + 0.174·19-s + 0.322·20-s − 0.502·21-s − 1.84·22-s + 0.644·23-s − 0.487·24-s + 0.200·25-s − 1.98·26-s − 0.308·27-s − 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 11 | \( 1 + 4.66T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 0.760T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.0265T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.07T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−7.44366160493443410594934074423, −6.87553810433619643083172656195, −5.73412158033153699430137211456, −5.33118669854918286140459301182, −4.70400635566870274094878230617, −3.72322216022798832459206688759, −3.07649631924495448695528665056, −2.62587124152631569264451449117, −1.93608863632132523570125612059, 0,
1.93608863632132523570125612059, 2.62587124152631569264451449117, 3.07649631924495448695528665056, 3.72322216022798832459206688759, 4.70400635566870274094878230617, 5.33118669854918286140459301182, 5.73412158033153699430137211456, 6.87553810433619643083172656195, 7.44366160493443410594934074423
