| L(s) = 1 | + 2.05e14·3-s + 1.15e18·4-s + 1.80e25·7-s + 4.23e28·9-s + 2.37e32·12-s − 2.12e33·13-s + 1.32e36·16-s + 3.76e38·19-s + 3.71e39·21-s + 8.67e41·25-s + 8.72e42·27-s + 2.08e43·28-s − 1.09e45·31-s + 4.88e46·36-s + 4.96e46·37-s − 4.36e47·39-s − 1.88e48·43-s + 2.73e50·48-s − 1.82e50·49-s − 2.44e51·52-s + 7.75e52·57-s − 5.54e53·61-s + 7.65e53·63-s + 1.53e54·64-s + 1.18e55·67-s + 1.55e56·73-s + 1.78e56·75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 0.800·7-s + 9-s + 12-s − 0.809·13-s + 16-s + 1.63·19-s + 0.800·21-s + 25-s + 27-s + 0.800·28-s − 1.99·31-s + 36-s + 0.446·37-s − 0.809·39-s − 0.186·43-s + 48-s − 0.358·49-s − 0.809·52-s + 1.63·57-s − 1.52·61-s + 0.800·63-s + 64-s + 1.94·67-s + 1.96·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(61-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+30) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{61}{2})\) |
\(\approx\) |
\(4.879088617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.879088617\) |
| \(L(31)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{30} T \) |
| good | 2 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 5 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 7 | \( 1 - \)\(18\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 11 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 13 | \( 1 + \)\(21\!\cdots\!02\)\( T + p^{60} T^{2} \) |
| 17 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 19 | \( 1 - \)\(37\!\cdots\!02\)\( T + p^{60} T^{2} \) |
| 23 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 29 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 31 | \( 1 + \)\(10\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 37 | \( 1 - \)\(49\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 41 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 43 | \( 1 + \)\(18\!\cdots\!02\)\( T + p^{60} T^{2} \) |
| 47 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 53 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 59 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 61 | \( 1 + \)\(55\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 67 | \( 1 - \)\(11\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 71 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 73 | \( 1 - \)\(15\!\cdots\!98\)\( T + p^{60} T^{2} \) |
| 79 | \( 1 - \)\(84\!\cdots\!02\)\( T + p^{60} T^{2} \) |
| 83 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 89 | \( ( 1 - p^{30} T )( 1 + p^{30} T ) \) |
| 97 | \( 1 + \)\(76\!\cdots\!02\)\( T + p^{60} T^{2} \) |
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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
−14.20507212151238455955019217469, −12.45101365216459997886079319105, −11.03154796668095194075847752190, −9.528301882883704609145086564990, −7.923997626609794099468509891656, −7.06232023904301993132064614572, −5.12824783853328385366928289524, −3.39510238351791712863472975331, −2.26883679062018412403233497953, −1.22828318029797474545519372882,
1.22828318029797474545519372882, 2.26883679062018412403233497953, 3.39510238351791712863472975331, 5.12824783853328385366928289524, 7.06232023904301993132064614572, 7.923997626609794099468509891656, 9.528301882883704609145086564990, 11.03154796668095194075847752190, 12.45101365216459997886079319105, 14.20507212151238455955019217469
