L(s) = 1 | + 2.12·2-s + 2.51·4-s + 2.07·5-s + 4.84·7-s + 1.09·8-s + 4.40·10-s − 4.14·11-s − 1.21·13-s + 10.3·14-s − 2.70·16-s − 2.36·17-s − 1.83·19-s + 5.20·20-s − 8.81·22-s + 4.30·23-s − 0.711·25-s − 2.58·26-s + 12.1·28-s − 2.98·29-s + 1.47·31-s − 7.93·32-s − 5.02·34-s + 10.0·35-s + 8.97·37-s − 3.90·38-s + 2.26·40-s − 2.26·41-s + ⋯ |
L(s) = 1 | + 1.50·2-s + 1.25·4-s + 0.926·5-s + 1.83·7-s + 0.387·8-s + 1.39·10-s − 1.25·11-s − 0.337·13-s + 2.75·14-s − 0.675·16-s − 0.573·17-s − 0.421·19-s + 1.16·20-s − 1.87·22-s + 0.897·23-s − 0.142·25-s − 0.506·26-s + 2.30·28-s − 0.553·29-s + 0.264·31-s − 1.40·32-s − 0.861·34-s + 1.69·35-s + 1.47·37-s − 0.633·38-s + 0.358·40-s − 0.353·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.049459232\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.049459232\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.12T + 2T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 - 4.84T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 8.97T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 0.262T + 59T^{2} \) |
| 61 | \( 1 + 4.45T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 5.10T + 97T^{2} \) |
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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
−10.82750832380592657856312433000, −9.669767239281719153548131582813, −8.517386611967797736160404079658, −7.67391846222819376325202448823, −6.56363574793811812675857002404, −5.45891608203138155470317072219, −5.07240677257135440828915794010, −4.25940095032795052966740600149, −2.69280749121232673769795368865, −1.90226317428666647700646517728,
1.90226317428666647700646517728, 2.69280749121232673769795368865, 4.25940095032795052966740600149, 5.07240677257135440828915794010, 5.45891608203138155470317072219, 6.56363574793811812675857002404, 7.67391846222819376325202448823, 8.517386611967797736160404079658, 9.669767239281719153548131582813, 10.82750832380592657856312433000