| L(s) = 1 | + 9.29e11·2-s − 3.97e24·4-s − 4.45e34·7-s − 8.18e36·8-s + 1.33e39·9-s + 9.42e42·11-s − 4.14e46·14-s + 1.15e49·16-s + 1.23e51·18-s + 8.75e54·22-s − 9.48e55·23-s + 2.06e57·25-s + 1.76e59·28-s + 1.67e60·29-s + 5.03e61·32-s − 5.28e63·36-s + 4.70e63·37-s − 1.64e67·43-s − 3.74e67·44-s − 8.82e67·46-s + 1.98e69·49-s + 1.92e69·50-s − 7.24e70·53-s + 3.64e71·56-s + 1.56e72·58-s − 5.92e73·63-s − 9.20e72·64-s + ⋯ |
| L(s) = 1 | + 0.422·2-s − 0.821·4-s − 7-s − 0.770·8-s + 9-s + 1.89·11-s − 0.422·14-s + 0.495·16-s + 0.422·18-s + 0.800·22-s − 1.40·23-s + 25-s + 0.821·28-s + 1.84·29-s + 0.979·32-s − 0.821·36-s + 0.237·37-s − 1.75·43-s − 1.55·44-s − 0.592·46-s + 49-s + 0.422·50-s − 1.46·53-s + 0.770·56-s + 0.781·58-s − 63-s − 0.0813·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(83-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+41) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{83}{2})\) |
\(\approx\) |
\(2.332743248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.332743248\) |
| \(L(42)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{41} T \) |
| good | 2 | \( 1 - 929828440857 T + p^{82} T^{2} \) |
| 3 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 5 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 11 | \( 1 - \)\(94\!\cdots\!34\)\( T + p^{82} T^{2} \) |
| 13 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 17 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 19 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 23 | \( 1 + \)\(94\!\cdots\!22\)\( T + p^{82} T^{2} \) |
| 29 | \( 1 - \)\(16\!\cdots\!46\)\( T + p^{82} T^{2} \) |
| 31 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 37 | \( 1 - \)\(47\!\cdots\!62\)\( T + p^{82} T^{2} \) |
| 41 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 43 | \( 1 + \)\(16\!\cdots\!22\)\( T + p^{82} T^{2} \) |
| 47 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 53 | \( 1 + \)\(72\!\cdots\!06\)\( T + p^{82} T^{2} \) |
| 59 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 61 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 67 | \( 1 + \)\(26\!\cdots\!98\)\( T + p^{82} T^{2} \) |
| 71 | \( 1 - \)\(78\!\cdots\!94\)\( T + p^{82} T^{2} \) |
| 73 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 79 | \( 1 + \)\(10\!\cdots\!14\)\( T + p^{82} T^{2} \) |
| 83 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 89 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
| 97 | \( ( 1 - p^{41} T )( 1 + p^{41} T ) \) |
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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
−10.11209714959250904421956072934, −9.458163908204503492362739067397, −8.442487802088475622034344504841, −6.79331729994480751369895002080, −6.18911295319627165238071374858, −4.67499342146127929769933151833, −3.96221274098823545532108198199, −3.12655742372242018231172580272, −1.50419697945957726754840088197, −0.59894253748706376839805072082,
0.59894253748706376839805072082, 1.50419697945957726754840088197, 3.12655742372242018231172580272, 3.96221274098823545532108198199, 4.67499342146127929769933151833, 6.18911295319627165238071374858, 6.79331729994480751369895002080, 8.442487802088475622034344504841, 9.458163908204503492362739067397, 10.11209714959250904421956072934
