| L(s) = 1 | − 9·3-s − 37·7-s + 54·9-s − 51·19-s + 333·21-s + 125·25-s − 243·27-s + 597·31-s − 433·37-s + 898·43-s + 1.02e3·49-s + 459·57-s + 1.62e3·61-s − 1.99e3·63-s + 1.00e3·67-s + 1.46e3·73-s − 1.12e3·75-s + 503·79-s + 729·81-s − 5.37e3·93-s − 3.62e3·103-s − 1.56e3·109-s + 3.89e3·111-s − 1.33e3·121-s + 127-s − 8.08e3·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.99·7-s + 2·9-s − 0.615·19-s + 3.46·21-s + 25-s − 1.73·27-s + 3.45·31-s − 1.92·37-s + 3.18·43-s + 2.99·49-s + 1.06·57-s + 3.40·61-s − 3.99·63-s + 1.83·67-s + 2.34·73-s − 1.73·75-s + 0.716·79-s + 81-s − 5.99·93-s − 3.46·103-s − 1.37·109-s + 3.33·111-s − 121-s + 0.000698·127-s − 5.51·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.107111573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107111573\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 7 | $C_2$ | \( 1 + 37 T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 89 T + p^{3} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 107 T + p^{3} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 - 289 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )( 1 + 323 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 449 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 901 T + p^{3} T^{2} )( 1 - 719 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )( 1 - 127 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )( 1 - 271 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 1387 T + p^{3} T^{2} )( 1 + 884 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14130036752018570764271435163, −11.02259997269818650279176482891, −10.30652703122725242881287287461, −10.20626143360082043177996149314, −9.632558061399048382202858460953, −9.306090616853137317116967001032, −8.510457380055838500842834489346, −8.124328974441269308574604807868, −7.09175492648458762585092338285, −6.87550363485240104444509645316, −6.39842052810271904728139066701, −6.24906047358422496304166192163, −5.32835139636629463882648038406, −5.23314685260740669038206188835, −4.13375822720980848253973000194, −3.99152180790329189962960505305, −2.93573807023199543799187689063, −2.36662635876735189927286967762, −0.74821054782764142460497242100, −0.68932070557933312828176033287,
0.68932070557933312828176033287, 0.74821054782764142460497242100, 2.36662635876735189927286967762, 2.93573807023199543799187689063, 3.99152180790329189962960505305, 4.13375822720980848253973000194, 5.23314685260740669038206188835, 5.32835139636629463882648038406, 6.24906047358422496304166192163, 6.39842052810271904728139066701, 6.87550363485240104444509645316, 7.09175492648458762585092338285, 8.124328974441269308574604807868, 8.510457380055838500842834489346, 9.306090616853137317116967001032, 9.632558061399048382202858460953, 10.20626143360082043177996149314, 10.30652703122725242881287287461, 11.02259997269818650279176482891, 11.14130036752018570764271435163
