| L(s) = 1 | + 0.186·3-s − 2.25·5-s − 3.52·7-s − 2.96·9-s − 0.515·11-s − 5.38·13-s − 0.419·15-s − 4.16·17-s − 4.92·19-s − 0.657·21-s + 7.69·23-s + 0.0674·25-s − 1.11·27-s − 8.93·29-s + 4.43·31-s − 0.0959·33-s + 7.94·35-s + 5.99·37-s − 1.00·39-s − 8.99·41-s + 1.66·43-s + 6.67·45-s − 8.55·47-s + 5.45·49-s − 0.775·51-s + 13.1·53-s + 1.16·55-s + ⋯ |
| L(s) = 1 | + 0.107·3-s − 1.00·5-s − 1.33·7-s − 0.988·9-s − 0.155·11-s − 1.49·13-s − 0.108·15-s − 1.01·17-s − 1.13·19-s − 0.143·21-s + 1.60·23-s + 0.0134·25-s − 0.213·27-s − 1.65·29-s + 0.795·31-s − 0.0167·33-s + 1.34·35-s + 0.986·37-s − 0.160·39-s − 1.40·41-s + 0.253·43-s + 0.995·45-s − 1.24·47-s + 0.779·49-s − 0.108·51-s + 1.80·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2160127358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2160127358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 3 | \( 1 - 0.186T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 0.515T + 11T^{2} \) |
| 13 | \( 1 + 5.38T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 5.99T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 9.25T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.443662851400905285565352722927, −7.78924667612108685557570559685, −6.92054826937034930188697811569, −6.50864267557884057942451414205, −5.45460403572757576905784599126, −4.64416855614159019784360602480, −3.77101918009902576080648125292, −2.97508479057407776509762584635, −2.30732189803234073073627095478, −0.24318795063137388496292229435,
0.24318795063137388496292229435, 2.30732189803234073073627095478, 2.97508479057407776509762584635, 3.77101918009902576080648125292, 4.64416855614159019784360602480, 5.45460403572757576905784599126, 6.50864267557884057942451414205, 6.92054826937034930188697811569, 7.78924667612108685557570559685, 8.443662851400905285565352722927
