| L(s) = 1 | + 1.71·2-s + 0.924·4-s − 1.37·5-s − 0.310·7-s − 1.83·8-s − 2.34·10-s + 2.41·11-s − 4.61·13-s − 0.530·14-s − 4.99·16-s − 0.210·17-s + 3.48·19-s − 1.26·20-s + 4.12·22-s + 8.00·23-s − 3.11·25-s − 7.89·26-s − 0.286·28-s − 8.31·29-s + 0.801·31-s − 4.86·32-s − 0.360·34-s + 0.426·35-s + 10.8·37-s + 5.95·38-s + 2.52·40-s − 4.69·41-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.462·4-s − 0.614·5-s − 0.117·7-s − 0.650·8-s − 0.742·10-s + 0.726·11-s − 1.28·13-s − 0.141·14-s − 1.24·16-s − 0.0510·17-s + 0.799·19-s − 0.283·20-s + 0.878·22-s + 1.67·23-s − 0.622·25-s − 1.54·26-s − 0.0541·28-s − 1.54·29-s + 0.143·31-s − 0.859·32-s − 0.0617·34-s + 0.0720·35-s + 1.78·37-s + 0.966·38-s + 0.399·40-s − 0.733·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589893148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589893148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + 0.310T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 0.210T + 17T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 23 | \( 1 - 8.00T + 23T^{2} \) |
| 29 | \( 1 + 8.31T + 29T^{2} \) |
| 31 | \( 1 - 0.801T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 4.69T + 41T^{2} \) |
| 43 | \( 1 - 7.43T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 1.25T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 + 0.802T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−7.75501560520782914134360240201, −7.17228032463069355862628636313, −6.54788508377797245497009669061, −5.63051798078140608219677111944, −5.08603160959098173848379486265, −4.38015524411837906269006596855, −3.71594538846054055174250873679, −3.05558723501777327685340475840, −2.17505089655567172181432399283, −0.67354225359400631996040005372,
0.67354225359400631996040005372, 2.17505089655567172181432399283, 3.05558723501777327685340475840, 3.71594538846054055174250873679, 4.38015524411837906269006596855, 5.08603160959098173848379486265, 5.63051798078140608219677111944, 6.54788508377797245497009669061, 7.17228032463069355862628636313, 7.75501560520782914134360240201
