| L(s) = 1 | − 32·4-s + 768·16-s + 8.97e3·19-s + 1.35e3·25-s + 1.54e4·31-s + 6.66e4·49-s − 1.52e5·61-s − 1.63e4·64-s − 2.87e5·76-s + 8.26e4·79-s − 4.32e4·100-s + 2.32e5·109-s + 1.88e5·121-s − 4.93e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.27e5·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 5.70·19-s + 0.431·25-s + 2.88·31-s + 3.96·49-s − 5.25·61-s − 1/2·64-s − 5.70·76-s + 1.49·79-s − 0.431·100-s + 1.87·109-s + 1.16·121-s − 2.88·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 2.49·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.354974342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.354974342\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 54 p^{2} T^{2} + p^{10} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 33310 T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 94074 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 463990 T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 1480158 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2244 T + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 12622686 T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40800682 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3856 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 89804410 T^{2} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 139752402 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 265922422 T^{2} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 62680014 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 835077670 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 33744102 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38158 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 1387552678 T^{2} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 3331783438 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 669338414 T^{2} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 20664 T + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 1524711738 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 7604090994 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17140767490 T^{2} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.498903326805782198275335312272, −9.205932870945034186401280892059, −8.916600789721389439904686171511, −8.585108557934751357534168887544, −8.337301901034010247843513820181, −7.68777205261882449604796187271, −7.48108338752398510549047863362, −7.42935052000147810832021188294, −7.40937087700404019310935425726, −6.56177205645134132007395873152, −6.23537907048040493572176812156, −5.87281770836362848766213883395, −5.57371754511329426895388020272, −5.17420518450467020485557048730, −4.84666853966694760169539836303, −4.80214290459624339511612084588, −4.06187410745322057614455350292, −3.82278917493018015931466175042, −3.04908629534306851877573308216, −3.00084136083834037633935615253, −2.84862985790611325952166676234, −1.73517065368450214618121847495, −1.07946158728403703436482706208, −0.917146354564714072031251182862, −0.59229486588608178435438139420,
0.59229486588608178435438139420, 0.917146354564714072031251182862, 1.07946158728403703436482706208, 1.73517065368450214618121847495, 2.84862985790611325952166676234, 3.00084136083834037633935615253, 3.04908629534306851877573308216, 3.82278917493018015931466175042, 4.06187410745322057614455350292, 4.80214290459624339511612084588, 4.84666853966694760169539836303, 5.17420518450467020485557048730, 5.57371754511329426895388020272, 5.87281770836362848766213883395, 6.23537907048040493572176812156, 6.56177205645134132007395873152, 7.40937087700404019310935425726, 7.42935052000147810832021188294, 7.48108338752398510549047863362, 7.68777205261882449604796187271, 8.337301901034010247843513820181, 8.585108557934751357534168887544, 8.916600789721389439904686171511, 9.205932870945034186401280892059, 9.498903326805782198275335312272
