| L(s) = 1 | − 4·3-s + 4·9-s + 4·27-s − 24·31-s − 24·37-s + 24·41-s − 20·43-s + 4·49-s + 20·67-s + 24·71-s + 48·79-s − 10·81-s − 12·83-s + 24·89-s + 96·93-s − 60·107-s + 96·111-s + 20·121-s − 96·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s − 16·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4/3·9-s + 0.769·27-s − 4.31·31-s − 3.94·37-s + 3.74·41-s − 3.04·43-s + 4/7·49-s + 2.44·67-s + 2.84·71-s + 5.40·79-s − 1.11·81-s − 1.31·83-s + 2.54·89-s + 9.95·93-s − 5.80·107-s + 9.11·111-s + 1.81·121-s − 8.65·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1415957176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1415957176\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.69960107900960116719417169282, −6.68929986677692118510140706360, −6.13545737207515525797784285937, −6.05629870827409417482628573842, −5.62580782303062663980840493457, −5.60871129735776259200355235388, −5.52903270288770044722426606934, −5.19569184620023023857101114300, −5.03993600287336669442377053856, −4.97768922843258220212581866855, −4.87097588614781965947943590383, −4.26845184809000422698765713610, −3.83556444474130628999796501594, −3.76903046843412352088818617593, −3.61930928510390447413421861623, −3.58631853016481276828094017169, −3.11745652410939262085376224231, −2.66176257245725524621528595562, −2.39130949933524008626340830293, −1.98484866011011060993434982237, −1.81698050031540470105310980378, −1.66948858401971837384382269558, −0.870757098151957887564925037967, −0.63697300345061816541401076135, −0.13703573424083539679759682059,
0.13703573424083539679759682059, 0.63697300345061816541401076135, 0.870757098151957887564925037967, 1.66948858401971837384382269558, 1.81698050031540470105310980378, 1.98484866011011060993434982237, 2.39130949933524008626340830293, 2.66176257245725524621528595562, 3.11745652410939262085376224231, 3.58631853016481276828094017169, 3.61930928510390447413421861623, 3.76903046843412352088818617593, 3.83556444474130628999796501594, 4.26845184809000422698765713610, 4.87097588614781965947943590383, 4.97768922843258220212581866855, 5.03993600287336669442377053856, 5.19569184620023023857101114300, 5.52903270288770044722426606934, 5.60871129735776259200355235388, 5.62580782303062663980840493457, 6.05629870827409417482628573842, 6.13545737207515525797784285937, 6.68929986677692118510140706360, 6.69960107900960116719417169282
