L(s) = 1 | + 110·5-s + 290·9-s + 296·11-s + 4.44e3·19-s + 8.97e3·25-s + 540·29-s + 4.09e3·31-s − 4.79e3·41-s + 3.19e4·45-s + 8.65e3·49-s + 3.25e4·55-s − 7.94e4·59-s − 8.45e4·61-s + 8.49e3·71-s + 7.05e4·79-s + 2.50e4·81-s + 1.70e5·89-s + 4.88e5·95-s + 8.58e4·99-s − 8.59e3·101-s + 7.19e4·109-s − 2.56e5·121-s + 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 1.96·5-s + 1.19·9-s + 0.737·11-s + 2.82·19-s + 2.87·25-s + 0.119·29-s + 0.765·31-s − 0.445·41-s + 2.34·45-s + 0.514·49-s + 1.45·55-s − 2.97·59-s − 2.91·61-s + 0.200·71-s + 1.27·79-s + 0.424·81-s + 2.28·89-s + 5.55·95-s + 0.880·99-s − 0.0838·101-s + 0.580·109-s − 1.59·121-s + 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(5.096226876\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.096226876\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 22 p T + p^{5} T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 290 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8650 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 148 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 274730 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1354590 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11320170 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 270 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2048 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 119573530 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2398 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288754450 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 344584890 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 827605690 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 39740 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1669968610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3239892370 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 35280 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7103795010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 85210 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7720618690 T^{2} + p^{10} T^{4} \) |
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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
−13.58248784421289701729257868103, −13.50659965957190715628203683561, −12.56767908793262022207124289458, −12.19671509682654663964408871948, −11.59846747118026925394831266567, −10.67475989277613844932529892691, −10.27991311618157684912840832220, −9.723601451032213785933466091870, −9.246543034284442442540938365465, −9.051773555324886597155218867365, −7.70784665236488771735719467181, −7.34926583302124053202932681555, −6.37389826249377757064204605115, −6.16985539287195084585772662105, −5.12958350681591005540140504081, −4.79017797052126808819223439258, −3.50316828688008904089271131000, −2.66908325633065254032997145101, −1.43484698478187625651289244074, −1.22449497840952489148991991128,
1.22449497840952489148991991128, 1.43484698478187625651289244074, 2.66908325633065254032997145101, 3.50316828688008904089271131000, 4.79017797052126808819223439258, 5.12958350681591005540140504081, 6.16985539287195084585772662105, 6.37389826249377757064204605115, 7.34926583302124053202932681555, 7.70784665236488771735719467181, 9.051773555324886597155218867365, 9.246543034284442442540938365465, 9.723601451032213785933466091870, 10.27991311618157684912840832220, 10.67475989277613844932529892691, 11.59846747118026925394831266567, 12.19671509682654663964408871948, 12.56767908793262022207124289458, 13.50659965957190715628203683561, 13.58248784421289701729257868103