L(s) = 1 | + 2-s − 9·3-s + 5-s − 9·6-s − 7-s + 45·9-s + 10-s + 11-s − 14-s − 9·15-s + 45·18-s − 19-s + 9·21-s + 22-s + 23-s − 165·27-s − 9·30-s − 9·33-s − 35-s − 38-s + 41-s + 9·42-s − 43-s + 45·45-s + 46-s + 47-s + 53-s + ⋯ |
L(s) = 1 | + 2-s − 9·3-s + 5-s − 9·6-s − 7-s + 45·9-s + 10-s + 11-s − 14-s − 9·15-s + 45·18-s − 19-s + 9·21-s + 22-s + 23-s − 165·27-s − 9·30-s − 9·33-s − 35-s − 38-s + 41-s + 9·42-s − 43-s + 45·45-s + 46-s + 47-s + 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 557^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 557^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05876328271\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05876328271\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T )^{9} \) |
| 557 | \( ( 1 + T )^{9} \) |
good | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 5 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 7 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 11 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 13 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 17 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 19 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 23 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 29 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 31 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 37 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 41 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 43 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 47 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 53 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 59 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 61 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 67 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 71 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 73 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 79 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
| 83 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 89 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 97 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.00869535216336956319935601644, −3.88801000521698560103234006393, −3.88135750065285215189030247080, −3.83278602319135456493431987092, −3.67459827134676097920089417540, −3.66117185652938655987884260349, −3.20572420508609169572972314011, −3.04494633285290967538426472302, −3.01013153347103660235309929914, −2.98000694871258629850290422272, −2.77298342848437064096016472808, −2.32235608023896616934057633168, −2.27860037132295437384101162753, −2.04244874444611431063975806443, −2.04014194154852307593356486324, −1.76851513362048831706608994184, −1.72958095942354866392015631615, −1.64871969398023675169113668130, −1.51717725844726943945485803639, −1.21243269892397948615240873489, −1.11298873160910866547329218775, −0.977323044950498296553514312375, −0.894943822898025597571574450536, −0.57502123343045510622356873323, −0.35493752612622000602068088784,
0.35493752612622000602068088784, 0.57502123343045510622356873323, 0.894943822898025597571574450536, 0.977323044950498296553514312375, 1.11298873160910866547329218775, 1.21243269892397948615240873489, 1.51717725844726943945485803639, 1.64871969398023675169113668130, 1.72958095942354866392015631615, 1.76851513362048831706608994184, 2.04014194154852307593356486324, 2.04244874444611431063975806443, 2.27860037132295437384101162753, 2.32235608023896616934057633168, 2.77298342848437064096016472808, 2.98000694871258629850290422272, 3.01013153347103660235309929914, 3.04494633285290967538426472302, 3.20572420508609169572972314011, 3.66117185652938655987884260349, 3.67459827134676097920089417540, 3.83278602319135456493431987092, 3.88135750065285215189030247080, 3.88801000521698560103234006393, 4.00869535216336956319935601644
Plot not available for L-functions of degree greater than 10.