| L(s) = 1 | − 3·3-s + 4·4-s + 13·7-s − 12·12-s − 23·13-s + 63·19-s − 39·21-s + 25·25-s + 27·27-s + 52·28-s + 105·31-s + 21·37-s + 69·39-s − 122·43-s + 120·49-s − 92·52-s − 189·57-s + 74·61-s − 64·64-s − 231·67-s − 189·73-s − 75·75-s + 252·76-s + 11·79-s − 81·81-s − 156·84-s − 299·91-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 13/7·7-s − 12-s − 1.76·13-s + 3.31·19-s − 1.85·21-s + 25-s + 27-s + 13/7·28-s + 3.38·31-s + 0.567·37-s + 1.76·39-s − 2.83·43-s + 2.44·49-s − 1.76·52-s − 3.31·57-s + 1.21·61-s − 64-s − 3.44·67-s − 2.58·73-s − 75-s + 3.31·76-s + 0.139·79-s − 81-s − 1.85·84-s − 3.28·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.420491545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.420491545\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 13 T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 23 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 - 26 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 - 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )( 1 + 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 109 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 143 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 131 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.65689978300750078958959491782, −11.56081295350907807233497157506, −11.46746260768984576946690613565, −10.46078382281036591346146206053, −10.06309336709990125137469583594, −9.990152877154331589706498133395, −8.931449116651797168455375099013, −8.539354784214062131765518135547, −7.72402823641262665726750255223, −7.54406925955680704454771264279, −7.08708094458024230804373366434, −6.46492699064690966013635163071, −5.83643310157735018030837081113, −5.17564955415740210701177982886, −4.73646538857552153626288163470, −4.69536704136162381246717991132, −2.89777553989860017832584097994, −2.86969478279402612168862771165, −1.61931980182889776829821582370, −0.904946675604415079418521654368,
0.904946675604415079418521654368, 1.61931980182889776829821582370, 2.86969478279402612168862771165, 2.89777553989860017832584097994, 4.69536704136162381246717991132, 4.73646538857552153626288163470, 5.17564955415740210701177982886, 5.83643310157735018030837081113, 6.46492699064690966013635163071, 7.08708094458024230804373366434, 7.54406925955680704454771264279, 7.72402823641262665726750255223, 8.539354784214062131765518135547, 8.931449116651797168455375099013, 9.990152877154331589706498133395, 10.06309336709990125137469583594, 10.46078382281036591346146206053, 11.46746260768984576946690613565, 11.56081295350907807233497157506, 11.65689978300750078958959491782
