| L(s) = 1 | − 4·3-s + 6·9-s − 16·17-s + 4·27-s − 24·47-s + 6·49-s + 64·51-s + 48·79-s − 37·81-s + 40·83-s − 8·109-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s − 24·147-s + 149-s + 151-s − 96·153-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s − 3.88·17-s + 0.769·27-s − 3.50·47-s + 6/7·49-s + 8.96·51-s + 5.40·79-s − 4.11·81-s + 4.39·83-s − 0.766·109-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s − 1.97·147-s + 0.0819·149-s + 0.0813·151-s − 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6154614481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6154614481\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.51530727164020920118128785748, −6.35367505677884087950366373125, −6.14952809283308466116788426607, −6.02717641917944624838808871033, −5.49810592099559209589744924158, −5.29537363513097536431220877157, −5.28620258211837422733964094934, −5.03463999652227522960404022742, −4.79951672695134963801350477065, −4.76552109722015745833079237049, −4.26499380482963040898820947108, −4.24726841395281649319856419949, −4.22424790489457997169987651325, −3.56622722195596193060140454625, −3.45110875732400806807329443058, −3.17486796212394515135986553542, −2.96691476679884335674545205561, −2.42269581131135541615673977431, −2.15627677858320513954708849280, −2.14776310040993072740362499879, −1.83415436008972393612715444535, −1.41748173979352115230614957211, −0.69804681015368937655032499766, −0.64197211265481564987827084951, −0.28731994301051079302390990365,
0.28731994301051079302390990365, 0.64197211265481564987827084951, 0.69804681015368937655032499766, 1.41748173979352115230614957211, 1.83415436008972393612715444535, 2.14776310040993072740362499879, 2.15627677858320513954708849280, 2.42269581131135541615673977431, 2.96691476679884335674545205561, 3.17486796212394515135986553542, 3.45110875732400806807329443058, 3.56622722195596193060140454625, 4.22424790489457997169987651325, 4.24726841395281649319856419949, 4.26499380482963040898820947108, 4.76552109722015745833079237049, 4.79951672695134963801350477065, 5.03463999652227522960404022742, 5.28620258211837422733964094934, 5.29537363513097536431220877157, 5.49810592099559209589744924158, 6.02717641917944624838808871033, 6.14952809283308466116788426607, 6.35367505677884087950366373125, 6.51530727164020920118128785748
