| L(s) = 1 | + 4·3-s + 6·9-s − 12·13-s − 16-s + 24·19-s − 4·27-s − 12·31-s + 20·37-s − 48·39-s − 4·48-s + 96·57-s + 8·61-s + 20·67-s + 24·73-s + 32·79-s − 37·81-s − 48·93-s + 40·97-s + 8·109-s + 80·111-s − 72·117-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 3.32·13-s − 1/4·16-s + 5.50·19-s − 0.769·27-s − 2.15·31-s + 3.28·37-s − 7.68·39-s − 0.577·48-s + 12.7·57-s + 1.02·61-s + 2.44·67-s + 2.80·73-s + 3.60·79-s − 4.11·81-s − 4.97·93-s + 4.06·97-s + 0.766·109-s + 7.59·111-s − 6.65·117-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.786628781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.786628781\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1054 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 3998 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 4322 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.139806958442382058356299822849, −7.88352141201085335872112352848, −7.70516523525033162017941841644, −7.42630613824105012773309796664, −7.39520917610058093275361626968, −7.31775538686263285416472727303, −7.06258620780537083217495270098, −6.52395162277621688711317129857, −6.09605847366919804460148865623, −5.99129514401020568825503883306, −5.40099739122978266624222503302, −5.24985340103808300097915671590, −4.97863658782715270985198834109, −4.93931939876387097826589277771, −4.74855835723566852369912650104, −3.75476446524374393398850826685, −3.71625485650797373136142951594, −3.47676681822620453935412557895, −3.46325832119581224401560127071, −2.61819772181064849607171230985, −2.59503202606996395796708432105, −2.39367046731321102593747668067, −2.19553677753245923300945769963, −1.28938405490221187358980306374, −0.76639050194924002143569331056,
0.76639050194924002143569331056, 1.28938405490221187358980306374, 2.19553677753245923300945769963, 2.39367046731321102593747668067, 2.59503202606996395796708432105, 2.61819772181064849607171230985, 3.46325832119581224401560127071, 3.47676681822620453935412557895, 3.71625485650797373136142951594, 3.75476446524374393398850826685, 4.74855835723566852369912650104, 4.93931939876387097826589277771, 4.97863658782715270985198834109, 5.24985340103808300097915671590, 5.40099739122978266624222503302, 5.99129514401020568825503883306, 6.09605847366919804460148865623, 6.52395162277621688711317129857, 7.06258620780537083217495270098, 7.31775538686263285416472727303, 7.39520917610058093275361626968, 7.42630613824105012773309796664, 7.70516523525033162017941841644, 7.88352141201085335872112352848, 8.139806958442382058356299822849
