L(s) = 1 | − 2.69·2-s + 5.24·4-s + 1.40·5-s − 7-s − 8.73·8-s − 3.78·10-s − 6.04·11-s + 13-s + 2.69·14-s + 13.0·16-s + 6.70·17-s + 1.53·19-s + 7.36·20-s + 16.2·22-s − 0.742·23-s − 3.02·25-s − 2.69·26-s − 5.24·28-s + 6.12·29-s + 10.0·31-s − 17.5·32-s − 18.0·34-s − 1.40·35-s − 6.49·37-s − 4.13·38-s − 12.2·40-s + 7.53·41-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 2.62·4-s + 0.628·5-s − 0.377·7-s − 3.08·8-s − 1.19·10-s − 1.82·11-s + 0.277·13-s + 0.719·14-s + 3.25·16-s + 1.62·17-s + 0.352·19-s + 1.64·20-s + 3.46·22-s − 0.154·23-s − 0.605·25-s − 0.527·26-s − 0.991·28-s + 1.13·29-s + 1.80·31-s − 3.10·32-s − 3.09·34-s − 0.237·35-s − 1.06·37-s − 0.670·38-s − 1.94·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6410777073\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6410777073\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 + 0.742T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 - 7.53T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 5.30T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 + 0.464T + 97T^{2} \) |
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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
−10.13190826856742547844294893756, −9.632936856549679211685860870237, −8.414767567539976128866358842667, −7.996469909535066831093271383204, −7.12241736237163997858095951060, −6.11112725695544905937468072707, −5.33969948472934010018883152722, −3.13428747617119946548362992930, −2.29619703724710862244551864603, −0.855520376933325142710778978618,
0.855520376933325142710778978618, 2.29619703724710862244551864603, 3.13428747617119946548362992930, 5.33969948472934010018883152722, 6.11112725695544905937468072707, 7.12241736237163997858095951060, 7.996469909535066831093271383204, 8.414767567539976128866358842667, 9.632936856549679211685860870237, 10.13190826856742547844294893756