L(s) = 1 | + 1.43e7·3-s − 5.36e8·4-s − 2.49e12·7-s + 1.37e14·9-s − 7.70e15·12-s − 3.92e18·19-s − 3.58e19·21-s + 1.86e20·25-s + 9.84e20·27-s + 1.34e21·28-s − 9.48e21·31-s − 7.36e22·36-s − 7.51e22·37-s + 1.82e24·43-s + 3.01e24·49-s − 5.63e25·57-s − 1.84e26·61-s − 3.42e26·63-s + 1.54e26·64-s − 3.51e26·67-s + 3.42e27·73-s + 2.67e27·75-s + 2.10e27·76-s − 6.39e27·79-s + 4.71e27·81-s + 1.92e28·84-s − 1.36e29·93-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 4-s − 1.39·7-s + 2·9-s − 1.73·12-s − 1.12·19-s − 2.41·21-s + 25-s + 1.73·27-s + 1.39·28-s − 2.24·31-s − 2·36-s − 1.37·37-s + 3.76·43-s + 0.936·49-s − 1.95·57-s − 2.38·61-s − 2.78·63-s + 64-s − 1.16·67-s + 3.28·73-s + 1.73·75-s + 1.12·76-s − 1.94·79-s + 81-s + 2.41·84-s − 3.89·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(15)\) |
\(\approx\) |
\(1.557665411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.557665411\) |
\(L(\frac{31}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63222986843902457616171516759, −12.44338508248290124982005653343, −10.85624882493359675891705533243, −10.55676556927689684389961847797, −9.558920341612446134062107517082, −9.318498562466087273289219318692, −8.865638993445539886912540829077, −8.492995520947261655588718357324, −7.45775076945606001523149816283, −7.20341929594517007233573798725, −6.32325682250160204156855606704, −5.61323995524880766182069813097, −4.57670981442294377327288041242, −4.18581397248839298245589176358, −3.48445742731489478000620941931, −3.18983948092927640249387278785, −2.35801538348285673396431960736, −1.94931114295699152049476924170, −0.979921407308752281584863247925, −0.26215381223237131854286688775,
0.26215381223237131854286688775, 0.979921407308752281584863247925, 1.94931114295699152049476924170, 2.35801538348285673396431960736, 3.18983948092927640249387278785, 3.48445742731489478000620941931, 4.18581397248839298245589176358, 4.57670981442294377327288041242, 5.61323995524880766182069813097, 6.32325682250160204156855606704, 7.20341929594517007233573798725, 7.45775076945606001523149816283, 8.492995520947261655588718357324, 8.865638993445539886912540829077, 9.318498562466087273289219318692, 9.558920341612446134062107517082, 10.55676556927689684389961847797, 10.85624882493359675891705533243, 12.44338508248290124982005653343, 12.63222986843902457616171516759