| L(s) = 1 | + 3-s − 2.92·5-s − 1.00·7-s + 9-s + 2.52·11-s − 0.482·13-s − 2.92·15-s − 3.13·17-s + 0.552·19-s − 1.00·21-s + 3.55·25-s + 27-s + 0.121·29-s + 8.23·31-s + 2.52·33-s + 2.92·35-s − 9.53·37-s − 0.482·39-s − 1.39·41-s + 6.20·43-s − 2.92·45-s + 8.69·47-s − 5.99·49-s − 3.13·51-s + 7.72·53-s − 7.37·55-s + 0.552·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.30·5-s − 0.378·7-s + 0.333·9-s + 0.759·11-s − 0.133·13-s − 0.754·15-s − 0.760·17-s + 0.126·19-s − 0.218·21-s + 0.710·25-s + 0.192·27-s + 0.0225·29-s + 1.47·31-s + 0.438·33-s + 0.494·35-s − 1.56·37-s − 0.0773·39-s − 0.217·41-s + 0.946·43-s − 0.435·45-s + 1.26·47-s − 0.856·49-s − 0.439·51-s + 1.06·53-s − 0.993·55-s + 0.0732·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 + 0.482T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 0.552T + 19T^{2} \) |
| 29 | \( 1 - 0.121T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 9.53T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 - 7.72T + 53T^{2} \) |
| 59 | \( 1 + 0.236T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 2.79T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 - 4.93T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.68518229800166854899368299903, −7.03943487443341027864828385185, −6.54891146180016248638412760253, −5.51575888650251721703543274653, −4.41255985551697746348123376964, −4.07733141437364154462062856636, −3.26861852791386354720409967506, −2.50264459610250946413562745189, −1.25305791167167643235654214654, 0,
1.25305791167167643235654214654, 2.50264459610250946413562745189, 3.26861852791386354720409967506, 4.07733141437364154462062856636, 4.41255985551697746348123376964, 5.51575888650251721703543274653, 6.54891146180016248638412760253, 7.03943487443341027864828385185, 7.68518229800166854899368299903
