| L(s) = 1 | − 4·3-s − 4-s + 10·9-s + 4·12-s + 16-s + 32·23-s − 20·27-s − 2·31-s − 10·36-s + 20·37-s − 12·47-s − 4·48-s − 21·49-s − 28·53-s + 16·59-s − 5·64-s − 10·67-s − 128·69-s + 20·71-s + 35·81-s − 24·89-s − 32·92-s + 8·93-s − 26·97-s − 8·103-s + 20·108-s − 80·111-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/2·4-s + 10/3·9-s + 1.15·12-s + 1/4·16-s + 6.67·23-s − 3.84·27-s − 0.359·31-s − 5/3·36-s + 3.28·37-s − 1.75·47-s − 0.577·48-s − 3·49-s − 3.84·53-s + 2.08·59-s − 5/8·64-s − 1.22·67-s − 15.4·69-s + 2.37·71-s + 35/9·81-s − 2.54·89-s − 3.33·92-s + 0.829·93-s − 2.63·97-s − 0.788·103-s + 1.92·108-s − 7.59·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.137424821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137424821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 3 p T^{2} + 200 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 6 T^{2} - 181 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T^{2} + 320 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 426 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_4\times C_2$ | \( 1 + 201 T^{2} + 17336 T^{4} + 201 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 134 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 270 T^{2} + 30179 T^{4} + 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 256 T^{2} + 28974 T^{4} + 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 13 T + 228 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.48369505572690581021023959685, −5.04421822636993312581915704086, −4.99672592731418112094011424665, −4.97253022725782291933388835925, −4.94886801440622726841952345873, −4.50333954128640190777380997918, −4.42089818161899160051462260949, −4.36402238360630483174122965702, −4.20302903358760696809289598581, −3.79014419383862595759232469853, −3.39515952321407790488912262864, −3.37930402626546639070943278359, −3.13784332752953905280868956757, −3.02849696394679924040842740299, −2.82517452002750547873990118415, −2.66578267326586907416951650500, −2.35146083172224955473885958233, −1.92417750604503088004660473592, −1.49516997226307554681809599781, −1.43197190776712929730520281370, −1.37218615691139770239634116864, −1.02568963644196181708013637154, −0.793502500621380926403010043009, −0.46687007040564884550808367053, −0.31297446947974179927242880302,
0.31297446947974179927242880302, 0.46687007040564884550808367053, 0.793502500621380926403010043009, 1.02568963644196181708013637154, 1.37218615691139770239634116864, 1.43197190776712929730520281370, 1.49516997226307554681809599781, 1.92417750604503088004660473592, 2.35146083172224955473885958233, 2.66578267326586907416951650500, 2.82517452002750547873990118415, 3.02849696394679924040842740299, 3.13784332752953905280868956757, 3.37930402626546639070943278359, 3.39515952321407790488912262864, 3.79014419383862595759232469853, 4.20302903358760696809289598581, 4.36402238360630483174122965702, 4.42089818161899160051462260949, 4.50333954128640190777380997918, 4.94886801440622726841952345873, 4.97253022725782291933388835925, 4.99672592731418112094011424665, 5.04421822636993312581915704086, 5.48369505572690581021023959685
