| L(s) = 1 | + 3.93e9·2-s + 1.08e19·4-s − 1.57e26·7-s + 2.45e28·8-s + 3.81e29·9-s + 2.91e32·11-s − 6.20e35·14-s + 4.65e37·16-s + 1.50e39·18-s + 1.14e42·22-s + 3.35e41·23-s + 2.16e43·25-s − 1.71e45·28-s − 2.38e45·29-s + 6.96e46·32-s + 4.14e48·36-s − 6.96e48·37-s + 7.08e50·43-s + 3.16e51·44-s + 1.32e51·46-s + 2.48e52·49-s + 8.52e52·50-s + 5.57e53·53-s − 3.87e54·56-s − 9.37e54·58-s − 6.01e55·63-s + 5.94e55·64-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.35·4-s − 7-s + 2.47·8-s + 9-s + 1.51·11-s − 1.83·14-s + 2.18·16-s + 1.83·18-s + 2.77·22-s + 0.205·23-s + 25-s − 2.35·28-s − 1.10·29-s + 1.52·32-s + 2.35·36-s − 1.69·37-s + 1.63·43-s + 3.57·44-s + 0.376·46-s + 49-s + 1.83·50-s + 1.96·53-s − 2.47·56-s − 2.02·58-s − 63-s + 0.606·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(63-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+31) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{63}{2})\) |
\(\approx\) |
\(9.159696375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.159696375\) |
| \(L(32)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{31} T \) |
| good | 2 | \( 1 - 3932792553 T + p^{62} T^{2} \) |
| 3 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 5 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 11 | \( 1 - \)\(29\!\cdots\!94\)\( T + p^{62} T^{2} \) |
| 13 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 17 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 19 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 23 | \( 1 - \)\(33\!\cdots\!82\)\( T + p^{62} T^{2} \) |
| 29 | \( 1 + \)\(23\!\cdots\!54\)\( T + p^{62} T^{2} \) |
| 31 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 37 | \( 1 + \)\(69\!\cdots\!62\)\( T + p^{62} T^{2} \) |
| 41 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 43 | \( 1 - \)\(70\!\cdots\!42\)\( T + p^{62} T^{2} \) |
| 47 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 53 | \( 1 - \)\(55\!\cdots\!06\)\( T + p^{62} T^{2} \) |
| 59 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 61 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 67 | \( 1 - \)\(74\!\cdots\!18\)\( T + p^{62} T^{2} \) |
| 71 | \( 1 - \)\(28\!\cdots\!14\)\( T + p^{62} T^{2} \) |
| 73 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 79 | \( 1 - \)\(10\!\cdots\!06\)\( T + p^{62} T^{2} \) |
| 83 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 89 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 97 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
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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
−12.07322510995190391895885515981, −10.72568361698372758381620345557, −9.299096120252366647512242445858, −7.07747342280334843730060651400, −6.57079909215480945051628281358, −5.35814832475925087052659321099, −4.05715000399782633143013801802, −3.56553689192381943492560362696, −2.24317214511573201003239523294, −1.05268485899384966427458322576,
1.05268485899384966427458322576, 2.24317214511573201003239523294, 3.56553689192381943492560362696, 4.05715000399782633143013801802, 5.35814832475925087052659321099, 6.57079909215480945051628281358, 7.07747342280334843730060651400, 9.299096120252366647512242445858, 10.72568361698372758381620345557, 12.07322510995190391895885515981
