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
\(L(1)\) |
\(\approx\) |
\(0.7806665572\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7806665572\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 5.10T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32960146471482295547549750993, −9.399519915104074426750834595465, −8.530846781810323755344945509281, −7.932351499072007601958651054561, −7.48828064849208615372400843257, −6.30626323908466607170406035130, −4.77920327286385818733136360621, −3.99673268960831030580931426632, −2.08079925344298818040771693188, −0.988671212626980202574840408207,
0.988671212626980202574840408207, 2.08079925344298818040771693188, 3.99673268960831030580931426632, 4.77920327286385818733136360621, 6.30626323908466607170406035130, 7.48828064849208615372400843257, 7.932351499072007601958651054561, 8.530846781810323755344945509281, 9.399519915104074426750834595465, 10.32960146471482295547549750993