| L(s) = 1 | + 6·3-s + 5·4-s + 27·9-s + 30·12-s + 9·16-s + 2·25-s + 108·27-s + 135·36-s + 140·43-s + 54·48-s − 98·49-s + 140·61-s − 35·64-s + 12·75-s − 100·79-s + 405·81-s + 10·100-s + 540·108-s − 190·121-s + 127-s + 840·129-s + 131-s + 137-s + 139-s + 243·144-s − 588·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2·3-s + 5/4·4-s + 3·9-s + 5/2·12-s + 9/16·16-s + 2/25·25-s + 4·27-s + 15/4·36-s + 3.25·43-s + 9/8·48-s − 2·49-s + 2.29·61-s − 0.546·64-s + 4/25·75-s − 1.26·79-s + 5·81-s + 1/10·100-s + 5·108-s − 1.57·121-s + 0.00787·127-s + 6.51·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.68·144-s − 4·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.447206960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.447206960\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2930 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4370 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6910 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 3790 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13730 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9550 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−10.66331118737804545679310184769, −10.57427054969919914306995775564, −9.946200080514051560603675896944, −9.363733300864795204731036431179, −9.335631988580804130223938302220, −8.565998970253369224373685288419, −8.250515603794071510861430105927, −7.80006279943227643761319987106, −7.23929181415586344808140072165, −7.13865280829160488377881077072, −6.49288364096579496039598808252, −6.03926757957942475963635654178, −5.22936094179421936132812318093, −4.50515377806908664652305643466, −3.97892598414359865802346512297, −3.46898227821435044978512619728, −2.66011253190601928105403460610, −2.55978651201906952958567741014, −1.80815078783040512786575534573, −1.10602792829486760561275321189,
1.10602792829486760561275321189, 1.80815078783040512786575534573, 2.55978651201906952958567741014, 2.66011253190601928105403460610, 3.46898227821435044978512619728, 3.97892598414359865802346512297, 4.50515377806908664652305643466, 5.22936094179421936132812318093, 6.03926757957942475963635654178, 6.49288364096579496039598808252, 7.13865280829160488377881077072, 7.23929181415586344808140072165, 7.80006279943227643761319987106, 8.250515603794071510861430105927, 8.565998970253369224373685288419, 9.335631988580804130223938302220, 9.363733300864795204731036431179, 9.946200080514051560603675896944, 10.57427054969919914306995775564, 10.66331118737804545679310184769
