| L(s) = 1 | − 2·4-s − 4·5-s − 4·11-s + 3·16-s + 8·19-s + 8·20-s + 8·25-s + 8·29-s + 16·31-s − 16·41-s + 8·44-s + 16·49-s + 16·55-s − 16·61-s − 4·64-s − 24·71-s − 16·76-s − 12·80-s − 32·95-s − 16·100-s + 40·101-s + 16·109-s − 16·116-s + 10·121-s − 32·124-s − 20·125-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s − 1.20·11-s + 3/4·16-s + 1.83·19-s + 1.78·20-s + 8/5·25-s + 1.48·29-s + 2.87·31-s − 2.49·41-s + 1.20·44-s + 16/7·49-s + 2.15·55-s − 2.04·61-s − 1/2·64-s − 2.84·71-s − 1.83·76-s − 1.34·80-s − 3.28·95-s − 8/5·100-s + 3.98·101-s + 1.53·109-s − 1.48·116-s + 0.909·121-s − 2.87·124-s − 1.78·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.453387409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453387409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 30 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 2258 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4470 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 918 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 11378 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 32 T^{2} - 1902 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 4842 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14598 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 10550 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.23562196563367393777080795463, −7.02296491459631259702043750637, −6.90638322986724148189153725366, −6.36410100555077408218772037522, −6.26952871328789547177652247544, −6.08101481121066859963635376203, −5.77400561077571415366522020090, −5.41131814513715435466012438036, −5.15536053256032508595861906341, −5.13982778830083795830294504738, −4.77912171548074574194042147138, −4.47838920046899879291515283068, −4.39325863062804177155071164525, −4.16695850958277873105606528628, −4.05770896716843677965322498699, −3.22441928741127283517906450665, −3.16959299272220307437937167001, −3.15724977854281539098074873724, −3.13574055295550622152301527340, −2.53857473304517875820991082219, −2.05659549537335290958329904540, −1.66996064163866236094087998387, −1.08443343204493986299223128912, −0.64059080719054639764348669905, −0.50948005776602084837527228826,
0.50948005776602084837527228826, 0.64059080719054639764348669905, 1.08443343204493986299223128912, 1.66996064163866236094087998387, 2.05659549537335290958329904540, 2.53857473304517875820991082219, 3.13574055295550622152301527340, 3.15724977854281539098074873724, 3.16959299272220307437937167001, 3.22441928741127283517906450665, 4.05770896716843677965322498699, 4.16695850958277873105606528628, 4.39325863062804177155071164525, 4.47838920046899879291515283068, 4.77912171548074574194042147138, 5.13982778830083795830294504738, 5.15536053256032508595861906341, 5.41131814513715435466012438036, 5.77400561077571415366522020090, 6.08101481121066859963635376203, 6.26952871328789547177652247544, 6.36410100555077408218772037522, 6.90638322986724148189153725366, 7.02296491459631259702043750637, 7.23562196563367393777080795463
