| L(s) = 1 | + 2.28e13·3-s − 5.91e23·7-s + 5.23e26·9-s − 3.09e31·13-s − 8.01e35·19-s − 1.35e37·21-s + 1.38e39·25-s + 1.19e40·27-s − 1.77e41·31-s − 1.31e44·37-s − 7.08e44·39-s + 6.26e45·43-s + 1.38e47·49-s − 1.83e49·57-s + 1.60e50·61-s − 3.09e50·63-s + 2.18e51·67-s − 2.68e52·73-s + 3.17e52·75-s + 2.69e53·79-s + 2.73e53·81-s + 1.83e55·91-s − 4.05e54·93-s + 8.17e55·97-s − 4.50e56·103-s + 1.36e57·109-s − 3.00e57·111-s + ⋯ |
| L(s) = 1 | + 3-s − 1.28·7-s + 9-s − 1.99·13-s − 1.25·19-s − 1.28·21-s + 25-s + 27-s − 0.309·31-s − 1.61·37-s − 1.99·39-s + 1.14·43-s + 0.654·49-s − 1.25·57-s + 1.64·61-s − 1.28·63-s + 1.62·67-s − 1.80·73-s + 75-s + 1.97·79-s + 81-s + 2.57·91-s − 0.309·93-s + 1.91·97-s − 1.96·103-s + 1.22·109-s − 1.61·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(57-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+28) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{57}{2})\) |
\(\approx\) |
\(1.835278020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835278020\) |
| \(L(29)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{28} T \) |
| good | 5 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 7 | \( 1 + \)\(59\!\cdots\!06\)\( T + p^{56} T^{2} \) |
| 11 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 13 | \( 1 + \)\(30\!\cdots\!86\)\( T + p^{56} T^{2} \) |
| 17 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 19 | \( 1 + \)\(80\!\cdots\!26\)\( T + p^{56} T^{2} \) |
| 23 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 29 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 31 | \( 1 + \)\(17\!\cdots\!86\)\( T + p^{56} T^{2} \) |
| 37 | \( 1 + \)\(13\!\cdots\!58\)\( T + p^{56} T^{2} \) |
| 41 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 43 | \( 1 - \)\(62\!\cdots\!14\)\( T + p^{56} T^{2} \) |
| 47 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 53 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 59 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 61 | \( 1 - \)\(16\!\cdots\!74\)\( T + p^{56} T^{2} \) |
| 67 | \( 1 - \)\(21\!\cdots\!14\)\( T + p^{56} T^{2} \) |
| 71 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 73 | \( 1 + \)\(26\!\cdots\!38\)\( T + p^{56} T^{2} \) |
| 79 | \( 1 - \)\(26\!\cdots\!22\)\( T + p^{56} T^{2} \) |
| 83 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 89 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 97 | \( 1 - \)\(81\!\cdots\!94\)\( T + p^{56} T^{2} \) |
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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
−10.41727313878195457943369507395, −9.622871039415271766961475267986, −8.701149840426586734402731132543, −7.35671542693324088628310059474, −6.61269842710151794809473414179, −4.98012259666351428478765194201, −3.81614104326135046685251717414, −2.79081518643645464823804293959, −2.08756921431930975367124153110, −0.48862803247704406718801448219,
0.48862803247704406718801448219, 2.08756921431930975367124153110, 2.79081518643645464823804293959, 3.81614104326135046685251717414, 4.98012259666351428478765194201, 6.61269842710151794809473414179, 7.35671542693324088628310059474, 8.701149840426586734402731132543, 9.622871039415271766961475267986, 10.41727313878195457943369507395
