| L(s) = 1 | + 6·5-s − 9-s − 18·17-s + 11·25-s − 14·37-s − 6·45-s − 6·53-s − 6·61-s + 18·73-s + 9·81-s − 108·85-s − 6·89-s − 18·101-s + 14·109-s + 48·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 1/3·9-s − 4.36·17-s + 11/5·25-s − 2.30·37-s − 0.894·45-s − 0.824·53-s − 0.768·61-s + 2.10·73-s + 81-s − 11.7·85-s − 0.635·89-s − 1.79·101-s + 1.34·109-s + 4.51·113-s − 0.0909·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.45·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059178603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059178603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + T^{2} - 120 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 31 T^{2} + 600 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 31 T^{2} - 1248 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 113 T^{2} + 8280 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 137 T^{2} + 12528 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.37253426368416677063296260948, −6.88676255904892415173507813346, −6.80732975759527254443882818079, −6.78061350528979505725629745195, −6.38679710477045818147983033576, −6.18056397595386981578727254036, −6.12802479011854783960056878058, −5.86239574785644351083548208802, −5.45913662093997898093211701397, −5.44286991589503904608586692744, −4.89774285576067412717145986331, −4.89766465455579452480983996719, −4.67177857677793635369546240818, −4.32123152525935443566052006544, −4.02137749741887733056354672407, −3.75517667811039268738461202582, −3.22709230755673541171014403680, −3.19971689517222703145765195368, −2.46950545956728809339955936577, −2.40879151630527325229761815194, −2.10044089283890058358601052338, −1.93468072503864251053803432649, −1.76393301916787636185043154828, −1.17664737245856564046973385862, −0.23108006771285037185584525750,
0.23108006771285037185584525750, 1.17664737245856564046973385862, 1.76393301916787636185043154828, 1.93468072503864251053803432649, 2.10044089283890058358601052338, 2.40879151630527325229761815194, 2.46950545956728809339955936577, 3.19971689517222703145765195368, 3.22709230755673541171014403680, 3.75517667811039268738461202582, 4.02137749741887733056354672407, 4.32123152525935443566052006544, 4.67177857677793635369546240818, 4.89766465455579452480983996719, 4.89774285576067412717145986331, 5.44286991589503904608586692744, 5.45913662093997898093211701397, 5.86239574785644351083548208802, 6.12802479011854783960056878058, 6.18056397595386981578727254036, 6.38679710477045818147983033576, 6.78061350528979505725629745195, 6.80732975759527254443882818079, 6.88676255904892415173507813346, 7.37253426368416677063296260948
