L(s) = 1 | − 1.67e3·4-s − 5.90e4·9-s + 1.75e6·16-s + 5.38e5·19-s + 1.83e7·31-s + 9.87e7·36-s − 5.64e8·49-s − 9.56e8·61-s − 1.17e9·64-s − 9.01e8·76-s + 2.49e9·79-s + 3.48e9·81-s + 2.98e10·109-s + 5.18e10·121-s − 3.07e10·124-s + 127-s + 131-s + 137-s + 139-s − 1.03e11·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.75e11·169-s − 3.18e10·171-s + ⋯ |
L(s) = 1 | − 1.63·4-s − 9-s + 1.66·16-s + 0.217·19-s + 0.642·31-s + 1.63·36-s − 2·49-s − 1.13·61-s − 1.09·64-s − 0.355·76-s + 0.809·79-s + 81-s + 1.93·109-s + 2·121-s − 1.04·124-s − 1.66·144-s − 2·169-s − 0.217·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.7734410768\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7734410768\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + 1673 T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1820608576898 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 269302 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 37064501443298 T^{2} + p^{20} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9196802 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7974026877035902 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 349504859353650098 T^{2} + p^{20} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 478013398 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1245148702 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 24050685636765349102 T^{2} + p^{20} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.40634153356000183780260256540, −12.35913339628007763874364282713, −11.60675574808179718661519385629, −11.16247802738918709730385972894, −10.35913092711627900610380888434, −9.865386149184748202700703025236, −9.200336136323217186527625170238, −8.943429745803582208187813720642, −8.051276607567053867346822178494, −8.045338379887344344576698813379, −6.90862347512091028919420224157, −6.09362319765777373339286658187, −5.54939855233053811236067185948, −4.82034950196919165057582376752, −4.44781364807369050440642799883, −3.48324437731948246140606225243, −3.04758344892759941730045872089, −1.95614075032671962511478539987, −0.946230189600472417003842597422, −0.30766867094748358602889004665,
0.30766867094748358602889004665, 0.946230189600472417003842597422, 1.95614075032671962511478539987, 3.04758344892759941730045872089, 3.48324437731948246140606225243, 4.44781364807369050440642799883, 4.82034950196919165057582376752, 5.54939855233053811236067185948, 6.09362319765777373339286658187, 6.90862347512091028919420224157, 8.045338379887344344576698813379, 8.051276607567053867346822178494, 8.943429745803582208187813720642, 9.200336136323217186527625170238, 9.865386149184748202700703025236, 10.35913092711627900610380888434, 11.16247802738918709730385972894, 11.60675574808179718661519385629, 12.35913339628007763874364282713, 13.40634153356000183780260256540