| L(s) = 1 | + 2.74e11·2-s − 2.54e18·3-s + 7.55e22·4-s − 6.98e29·6-s + 2.07e34·8-s + 4.62e36·9-s − 3.53e39·11-s − 1.91e41·12-s + 5.70e45·16-s + 7.06e46·17-s + 1.27e48·18-s + 2.41e48·19-s − 9.71e50·22-s − 5.27e52·24-s + 1.32e53·25-s − 7.12e54·27-s + 1.56e57·32-s + 8.97e57·33-s + 1.94e58·34-s + 3.49e59·36-s + 6.63e59·38-s + 3.09e61·41-s − 2.35e62·43-s − 2.67e62·44-s − 1.45e64·48-s + 1.68e64·49-s + 3.63e64·50-s + ⋯ |
| L(s) = 1 | + 2-s − 1.88·3-s + 4-s − 1.88·6-s + 8-s + 2.53·9-s − 0.944·11-s − 1.88·12-s + 16-s + 1.23·17-s + 2.53·18-s + 0.616·19-s − 0.944·22-s − 1.88·24-s + 25-s − 2.88·27-s + 32-s + 1.77·33-s + 1.23·34-s + 2.53·36-s + 0.616·38-s + 1.60·41-s − 1.99·43-s − 0.944·44-s − 1.88·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(77-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+38) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{77}{2})\) |
\(\approx\) |
\(2.625435783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625435783\) |
| \(L(39)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{38} T \) |
| good | 3 | \( 1 + 2540277181395969134 T + p^{76} T^{2} \) |
| 5 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 7 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 11 | \( 1 + \)\(35\!\cdots\!06\)\( T + p^{76} T^{2} \) |
| 13 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 17 | \( 1 - \)\(70\!\cdots\!26\)\( T + p^{76} T^{2} \) |
| 19 | \( 1 - \)\(24\!\cdots\!14\)\( T + p^{76} T^{2} \) |
| 23 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 29 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 31 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 37 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 41 | \( 1 - \)\(30\!\cdots\!94\)\( T + p^{76} T^{2} \) |
| 43 | \( 1 + \)\(23\!\cdots\!02\)\( T + p^{76} T^{2} \) |
| 47 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 53 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 59 | \( 1 - \)\(23\!\cdots\!42\)\( T + p^{76} T^{2} \) |
| 61 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 67 | \( 1 + \)\(41\!\cdots\!14\)\( T + p^{76} T^{2} \) |
| 71 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 73 | \( 1 + \)\(11\!\cdots\!54\)\( T + p^{76} T^{2} \) |
| 79 | \( ( 1 - p^{38} T )( 1 + p^{38} T ) \) |
| 83 | \( 1 - \)\(12\!\cdots\!66\)\( T + p^{76} T^{2} \) |
| 89 | \( 1 + \)\(97\!\cdots\!26\)\( T + p^{76} T^{2} \) |
| 97 | \( 1 + \)\(58\!\cdots\!22\)\( T + p^{76} T^{2} \) |
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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
−10.76104695258395002424559146181, −10.04758144099803861650869917895, −7.67001905116465591290189871688, −6.79373119916479414954488386255, −5.68350017122374110168891809228, −5.24761053242613352724544752894, −4.27678926040807510165383419525, −2.96783385970443304351558613613, −1.48937857957573819193590288743, −0.63521794969042916477380492158,
0.63521794969042916477380492158, 1.48937857957573819193590288743, 2.96783385970443304351558613613, 4.27678926040807510165383419525, 5.24761053242613352724544752894, 5.68350017122374110168891809228, 6.79373119916479414954488386255, 7.67001905116465591290189871688, 10.04758144099803861650869917895, 10.76104695258395002424559146181
