| L(s) = 1 | − 4·3-s + 4-s + 10·9-s − 4·12-s + 4·16-s − 6·17-s − 12·23-s − 14·25-s − 32·27-s − 6·29-s + 10·36-s + 16·43-s − 16·48-s + 14·49-s + 24·51-s − 12·53-s − 2·61-s + 11·64-s − 6·68-s + 48·69-s + 56·75-s + 16·79-s + 89·81-s + 24·87-s − 12·92-s − 14·100-s − 6·101-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s + 10/3·9-s − 1.15·12-s + 16-s − 1.45·17-s − 2.50·23-s − 2.79·25-s − 6.15·27-s − 1.11·29-s + 5/3·36-s + 2.43·43-s − 2.30·48-s + 2·49-s + 3.36·51-s − 1.64·53-s − 0.256·61-s + 11/8·64-s − 0.727·68-s + 5.77·69-s + 6.46·75-s + 1.80·79-s + 89/9·81-s + 2.57·87-s − 1.25·92-s − 7/5·100-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3697243870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3697243870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + T^{2} - 1368 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 143 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 - 146 T^{2} + 11907 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.349665201872342313588955562620, −9.293738423359049310958431943775, −9.199571952202992873146175998705, −8.276433748151106426137030070353, −8.138632461951013330312426057722, −8.108558979652241302615090395837, −7.45145844507463007729883656062, −7.34049854202209080245702459206, −7.25815811685853860469678321852, −6.96304453987406784930278395185, −6.15359377268972769751707187780, −6.03641800405607795597681235072, −5.91962893181555972810241465281, −5.86009954120082912247415771514, −5.78011015459741765677317590653, −4.92344190111112481042795812599, −4.82934182953175803873413901625, −4.34192001417676835383240037133, −3.91237810568063505552473766069, −3.75789500024280062620500480945, −3.55982381203645878803030763207, −2.16477055288425466478804862196, −2.16476351842961474470588166149, −1.80752134955969724882689588861, −0.48155172633364216074792975011,
0.48155172633364216074792975011, 1.80752134955969724882689588861, 2.16476351842961474470588166149, 2.16477055288425466478804862196, 3.55982381203645878803030763207, 3.75789500024280062620500480945, 3.91237810568063505552473766069, 4.34192001417676835383240037133, 4.82934182953175803873413901625, 4.92344190111112481042795812599, 5.78011015459741765677317590653, 5.86009954120082912247415771514, 5.91962893181555972810241465281, 6.03641800405607795597681235072, 6.15359377268972769751707187780, 6.96304453987406784930278395185, 7.25815811685853860469678321852, 7.34049854202209080245702459206, 7.45145844507463007729883656062, 8.108558979652241302615090395837, 8.138632461951013330312426057722, 8.276433748151106426137030070353, 9.199571952202992873146175998705, 9.293738423359049310958431943775, 9.349665201872342313588955562620
