| L(s) = 1 | + 360·3-s − 2.52e4·7-s + 6.48e4·9-s + 4.55e5·13-s + 2.49e5·16-s − 9.07e6·21-s + 7.08e6·27-s − 6.91e6·31-s − 2.30e7·37-s + 1.63e8·39-s + 9.74e7·43-s + 8.98e7·48-s + 3.17e8·49-s + 6.07e7·61-s − 1.63e9·63-s + 2.03e8·67-s − 1.59e8·73-s + 5.15e8·81-s − 1.14e10·91-s − 2.49e9·93-s + 3.22e9·97-s − 5.43e7·103-s − 8.28e9·111-s − 6.29e9·112-s + 2.94e10·117-s + 9.23e9·121-s + ⋯ |
| L(s) = 1 | + 2.56·3-s − 3.96·7-s + 3.29·9-s + 4.41·13-s + 0.952·16-s − 10.1·21-s + 2.56·27-s − 1.34·31-s − 2.01·37-s + 11.3·39-s + 4.34·43-s + 2.44·48-s + 7.86·49-s + 0.561·61-s − 13.0·63-s + 1.23·67-s − 0.656·73-s + 1.33·81-s − 17.5·91-s − 3.45·93-s + 3.69·97-s − 0.0475·103-s − 5.18·111-s − 3.77·112-s + 14.5·117-s + 3.91·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(11.46631452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.46631452\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2^2$ | \( 1 - 40 p^{2} T + 800 p^{4} T^{2} - 40 p^{11} T^{3} + p^{18} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 15607 p^{4} T^{4} + p^{36} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 1800 p T + 1620000 p^{2} T^{2} + 1800 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4615584982 T^{2} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 227520 T + 25882675200 T^{2} - 227520 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - \)\(15\!\cdots\!82\)\( T^{4} + p^{36} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 639840630742 T^{2} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - \)\(55\!\cdots\!62\)\( T^{4} + p^{36} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 2555497446662 T^{2} + p^{18} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1729928 T + p^{9} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 11507040 T + 66205984780800 T^{2} + 11507040 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 416558667805522 T^{2} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 48738600 T + 1187725564980000 T^{2} - 48738600 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + \)\(18\!\cdots\!78\)\( T^{4} + p^{36} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - \)\(10\!\cdots\!22\)\( T^{4} + p^{36} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 6491019052340278 T^{2} + p^{18} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 15186242 T + p^{9} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 101529720 T + 5154142021639200 T^{2} - 101529720 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 89874898731400462 T^{2} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 79621920 T + 3169825072243200 T^{2} + 79621920 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 200107037864981662 T^{2} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + \)\(69\!\cdots\!18\)\( T^{4} + p^{36} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 127978958256992818 T^{2} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1611067680 T + 1297769534770291200 T^{2} - 1611067680 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.941045928587696476888905865541, −8.659952376219734270524103250988, −8.432024281717743798892137249469, −8.125006243469581369970524844319, −7.62735186558161830453209733597, −7.22048333050381154778655699228, −7.11294782317142815492729240718, −6.67952607550220810267701544252, −6.22695763760732550286358078803, −6.00767126095503456428325951737, −5.83845732324428750553162390023, −5.79914832626585048608569438623, −4.81807836381641872279588835630, −3.89074543104492749411943234378, −3.87941348159985146041525376194, −3.65577712457708649882143707406, −3.51582456751910778018386734991, −3.22975873496161226765515612107, −2.94922055971680256793307857280, −2.54265314462445184281428026226, −2.06308324040034381686274468514, −1.61577998484982074619602119111, −0.996702500896695841509137153938, −0.796181334392890507614566217595, −0.37685010347195845201505918535,
0.37685010347195845201505918535, 0.796181334392890507614566217595, 0.996702500896695841509137153938, 1.61577998484982074619602119111, 2.06308324040034381686274468514, 2.54265314462445184281428026226, 2.94922055971680256793307857280, 3.22975873496161226765515612107, 3.51582456751910778018386734991, 3.65577712457708649882143707406, 3.87941348159985146041525376194, 3.89074543104492749411943234378, 4.81807836381641872279588835630, 5.79914832626585048608569438623, 5.83845732324428750553162390023, 6.00767126095503456428325951737, 6.22695763760732550286358078803, 6.67952607550220810267701544252, 7.11294782317142815492729240718, 7.22048333050381154778655699228, 7.62735186558161830453209733597, 8.125006243469581369970524844319, 8.432024281717743798892137249469, 8.659952376219734270524103250988, 8.941045928587696476888905865541
