| L(s) = 1 | + 12·23-s − 25-s − 12·41-s + 12·47-s − 14·49-s + 24·71-s + 4·73-s + 12·89-s + 20·97-s + 24·103-s − 24·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.50·23-s − 1/5·25-s − 1.87·41-s + 1.75·47-s − 2·49-s + 2.84·71-s + 0.468·73-s + 1.27·89-s + 2.03·97-s + 2.36·103-s − 2.25·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.726328690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.726328690\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.995023667265877932221464534425, −8.573091330886350399659647027354, −8.201738405221438610130321612715, −7.945330926207440916273343694046, −7.28739593104935365220407784665, −7.16258733738589617729740237835, −6.63781115949250136543882480737, −6.48922672055492464537477326866, −5.86713601413244357990765643181, −5.45350893226106825946899862817, −5.03821679825575861947731981153, −4.75029737616232500864283352058, −4.38637631019046432340169399509, −3.62777866060166136172783597331, −3.21835517779007736385053037554, −3.15150350402670343791280774122, −2.15290710932403608688646194148, −2.00814878055854722772814245621, −1.07744917579644821256237177127, −0.60935198174608818132866900175,
0.60935198174608818132866900175, 1.07744917579644821256237177127, 2.00814878055854722772814245621, 2.15290710932403608688646194148, 3.15150350402670343791280774122, 3.21835517779007736385053037554, 3.62777866060166136172783597331, 4.38637631019046432340169399509, 4.75029737616232500864283352058, 5.03821679825575861947731981153, 5.45350893226106825946899862817, 5.86713601413244357990765643181, 6.48922672055492464537477326866, 6.63781115949250136543882480737, 7.16258733738589617729740237835, 7.28739593104935365220407784665, 7.945330926207440916273343694046, 8.201738405221438610130321612715, 8.573091330886350399659647027354, 8.995023667265877932221464534425
