| L(s) = 1 | + 8·7-s + 12·17-s − 16·23-s − 25-s + 4·31-s + 8·41-s + 8·47-s + 34·49-s − 16·71-s + 20·73-s + 4·79-s − 8·89-s + 4·97-s + 16·103-s + 12·113-s + 96·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 2.91·17-s − 3.33·23-s − 1/5·25-s + 0.718·31-s + 1.24·41-s + 1.16·47-s + 34/7·49-s − 1.89·71-s + 2.34·73-s + 0.450·79-s − 0.847·89-s + 0.406·97-s + 1.57·103-s + 1.12·113-s + 8.80·119-s + 1.63·121-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 − 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.052993311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.052993311\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.732259326286372020326956608627, −9.472323453674279916113763927129, −8.682408339072633161691549019244, −8.345288538482188209427225127261, −7.961803585223090766698286915690, −7.959196371998677656810361217209, −7.44179592171566398057975556138, −7.29483271664638327520480785574, −6.21501859347869965512862742235, −5.87727677104827238558043301941, −5.61589460502288499767928858988, −5.21250004138566241553875619055, −4.54793714785498945648662448441, −4.42819655491698600847129325637, −3.79142535377333640068701407195, −3.35556596660228484826658126906, −2.32304563961746761523849879206, −2.06926274494542792983320061669, −1.39053384193613825922961368704, −0.925447556043918230638813365968,
0.925447556043918230638813365968, 1.39053384193613825922961368704, 2.06926274494542792983320061669, 2.32304563961746761523849879206, 3.35556596660228484826658126906, 3.79142535377333640068701407195, 4.42819655491698600847129325637, 4.54793714785498945648662448441, 5.21250004138566241553875619055, 5.61589460502288499767928858988, 5.87727677104827238558043301941, 6.21501859347869965512862742235, 7.29483271664638327520480785574, 7.44179592171566398057975556138, 7.959196371998677656810361217209, 7.961803585223090766698286915690, 8.345288538482188209427225127261, 8.682408339072633161691549019244, 9.472323453674279916113763927129, 9.732259326286372020326956608627
