| L(s) = 1 | − 1.65e13·2-s + 1.98e26·4-s − 2.18e36·7-s − 2.00e39·8-s + 1.07e41·9-s − 7.65e44·11-s + 3.62e49·14-s + 1.79e52·16-s − 1.78e54·18-s + 1.27e58·22-s + 7.16e58·23-s + 1.29e60·25-s − 4.32e62·28-s + 1.43e63·29-s − 1.42e65·32-s + 2.13e67·36-s − 5.04e67·37-s − 1.38e70·43-s − 1.51e71·44-s − 1.18e72·46-s + 4.76e72·49-s − 2.14e73·50-s + 1.80e74·53-s + 4.37e75·56-s − 2.37e76·58-s − 2.35e77·63-s + 9.74e77·64-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.55·4-s − 7-s − 2.94·8-s + 9-s − 1.27·11-s + 1.88·14-s + 2.99·16-s − 1.88·18-s + 2.39·22-s + 1.99·23-s + 25-s − 2.55·28-s + 1.87·29-s − 2.70·32-s + 2.55·36-s − 1.86·37-s − 0.800·43-s − 3.25·44-s − 3.77·46-s + 49-s − 1.88·50-s + 1.29·53-s + 2.94·56-s − 3.53·58-s − 63-s + 2.10·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(87-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+43) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{87}{2})\) |
\(\approx\) |
\(0.7835380972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7835380972\) |
| \(L(44)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{43} T \) |
| good | 2 | \( 1 + 16594750398327 T + p^{86} T^{2} \) |
| 3 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 5 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 11 | \( 1 + \)\(76\!\cdots\!66\)\( T + p^{86} T^{2} \) |
| 13 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 17 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 19 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 23 | \( 1 - \)\(71\!\cdots\!82\)\( T + p^{86} T^{2} \) |
| 29 | \( 1 - \)\(14\!\cdots\!86\)\( T + p^{86} T^{2} \) |
| 31 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 37 | \( 1 + \)\(50\!\cdots\!22\)\( T + p^{86} T^{2} \) |
| 41 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 43 | \( 1 + \)\(13\!\cdots\!38\)\( T + p^{86} T^{2} \) |
| 47 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 53 | \( 1 - \)\(18\!\cdots\!46\)\( T + p^{86} T^{2} \) |
| 59 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 61 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 67 | \( 1 - \)\(47\!\cdots\!18\)\( T + p^{86} T^{2} \) |
| 71 | \( 1 + \)\(55\!\cdots\!06\)\( T + p^{86} T^{2} \) |
| 73 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 79 | \( 1 - \)\(60\!\cdots\!66\)\( T + p^{86} T^{2} \) |
| 83 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 89 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
| 97 | \( ( 1 - p^{43} T )( 1 + p^{43} T ) \) |
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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
−10.27602298992906922017693793195, −9.203946837124601974708446225352, −8.321627697873489358079136048556, −7.08972856391424933368331530501, −6.70675605777696241884603003469, −5.11380176004971424634845215866, −3.20513351175073744203462288969, −2.47170347493852041227206331915, −1.21927386554905420586945832952, −0.50941444237654721225119794864,
0.50941444237654721225119794864, 1.21927386554905420586945832952, 2.47170347493852041227206331915, 3.20513351175073744203462288969, 5.11380176004971424634845215866, 6.70675605777696241884603003469, 7.08972856391424933368331530501, 8.321627697873489358079136048556, 9.203946837124601974708446225352, 10.27602298992906922017693793195
