| L(s) = 1 | − 2·3-s + 4-s − 5·11-s − 2·12-s + 3·13-s + 16-s − 17-s + 19-s + 7·25-s + 5·27-s − 7·31-s + 10·33-s − 6·37-s − 6·39-s − 5·44-s + 15·47-s − 2·48-s − 12·49-s + 2·51-s + 3·52-s − 2·57-s + 64-s − 9·67-s − 68-s + 13·71-s − 14·75-s + 76-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.50·11-s − 0.577·12-s + 0.832·13-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 7/5·25-s + 0.962·27-s − 1.25·31-s + 1.74·33-s − 0.986·37-s − 0.960·39-s − 0.753·44-s + 2.18·47-s − 0.288·48-s − 1.71·49-s + 0.280·51-s + 0.416·52-s − 0.264·57-s + 1/8·64-s − 1.09·67-s − 0.121·68-s + 1.54·71-s − 1.61·75-s + 0.114·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 157 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 149 T^{2} + p^{2} T^{4} \) |
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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
−8.539479686692923089414383310978, −8.121712939509453733992291657971, −7.63853028011648799382089432871, −7.03223482600329342373691365679, −6.72726419410680506040191819425, −6.16900890474979719543996852687, −5.59940836267347679606321059535, −5.37131741592415199123557227416, −4.93420255140770287688688052712, −4.20828305431104448874299692185, −3.40592343573803806106091479581, −2.88437731321729416407301810172, −2.19043427986223445872928802363, −1.17087525401043749137781251616, 0,
1.17087525401043749137781251616, 2.19043427986223445872928802363, 2.88437731321729416407301810172, 3.40592343573803806106091479581, 4.20828305431104448874299692185, 4.93420255140770287688688052712, 5.37131741592415199123557227416, 5.59940836267347679606321059535, 6.16900890474979719543996852687, 6.72726419410680506040191819425, 7.03223482600329342373691365679, 7.63853028011648799382089432871, 8.121712939509453733992291657971, 8.539479686692923089414383310978
