| L(s) = 1 | + 9·5-s − 31·7-s − 27·9-s − 15·11-s + 37·13-s − 84·17-s + 56·19-s + 195·23-s + 125·25-s − 111·29-s − 205·31-s − 279·35-s − 332·37-s + 261·41-s − 43·43-s − 243·45-s + 177·47-s + 343·49-s + 228·53-s − 135·55-s + 159·59-s − 191·61-s + 837·63-s + 333·65-s − 421·67-s − 312·71-s + 364·73-s + ⋯ |
| L(s) = 1 | + 0.804·5-s − 1.67·7-s − 9-s − 0.411·11-s + 0.789·13-s − 1.19·17-s + 0.676·19-s + 1.76·23-s + 25-s − 0.710·29-s − 1.18·31-s − 1.34·35-s − 1.47·37-s + 0.994·41-s − 0.152·43-s − 0.804·45-s + 0.549·47-s + 49-s + 0.590·53-s − 0.330·55-s + 0.350·59-s − 0.400·61-s + 1.67·63-s + 0.635·65-s − 0.767·67-s − 0.521·71-s + 0.583·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.368109392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368109392\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 31 T + 618 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 37 T - 828 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 195 T + 25858 T^{2} - 195 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 111 T - 12068 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 205 T + 12234 T^{2} + 205 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 261 T - 800 T^{2} - 261 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + p T - 42 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 177 T - 72494 T^{2} - 177 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 159 T - 180098 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 191 T - 190500 T^{2} + 191 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 421 T - 123522 T^{2} + 421 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 156 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1133 T + 790650 T^{2} - 1133 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1083 T + 601102 T^{2} + 1083 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1050 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 901 T - 100872 T^{2} - 901 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.90701471902972991806865397023, −12.63454411533842024513409871731, −11.93359629859719486312321824100, −11.08685161137955572248072520465, −10.93515162410398207770947964488, −10.44434004476588536725363606051, −9.535408448848577925756927779509, −9.387684353934934106698772709325, −8.792069137818794532315345192120, −8.483692024779758458543731784478, −7.29865575009638333516137325121, −6.93840614120262923114859104892, −6.39345413794908118157227988492, −5.57359751176674089642378312113, −5.51497934120753494774056330432, −4.37782043660692309743559654858, −3.20984528326290633803218854987, −3.09811646190709306470295566380, −1.99972156492406522762971089083, −0.56050723159954526261225446280,
0.56050723159954526261225446280, 1.99972156492406522762971089083, 3.09811646190709306470295566380, 3.20984528326290633803218854987, 4.37782043660692309743559654858, 5.51497934120753494774056330432, 5.57359751176674089642378312113, 6.39345413794908118157227988492, 6.93840614120262923114859104892, 7.29865575009638333516137325121, 8.483692024779758458543731784478, 8.792069137818794532315345192120, 9.387684353934934106698772709325, 9.535408448848577925756927779509, 10.44434004476588536725363606051, 10.93515162410398207770947964488, 11.08685161137955572248072520465, 11.93359629859719486312321824100, 12.63454411533842024513409871731, 12.90701471902972991806865397023
