| L(s) = 1 | + 3-s + 4·5-s − 9-s + 2·11-s + 13-s + 4·15-s − 12·19-s − 2·23-s + 2·25-s − 29-s + 11·31-s + 2·33-s − 12·37-s + 39-s − 41-s − 4·43-s − 4·45-s + 19·47-s − 20·53-s + 8·55-s − 12·57-s + 4·59-s + 4·65-s + 22·67-s − 2·69-s − 71-s + 9·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1/3·9-s + 0.603·11-s + 0.277·13-s + 1.03·15-s − 2.75·19-s − 0.417·23-s + 2/5·25-s − 0.185·29-s + 1.97·31-s + 0.348·33-s − 1.97·37-s + 0.160·39-s − 0.156·41-s − 0.609·43-s − 0.596·45-s + 2.77·47-s − 2.74·53-s + 1.07·55-s − 1.58·57-s + 0.520·59-s + 0.496·65-s + 2.68·67-s − 0.240·69-s − 0.118·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.304829663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.304829663\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11 T + 88 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 19 T + 180 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 238 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 138 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 128 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.913769290928375486450121330831, −7.79221960270476948666552507158, −7.07891161020720600668814371528, −6.65554246258793523144270584452, −6.47003123866795944021666941504, −6.25703806751738242539654939238, −6.04791937644171736171060806038, −5.48401149006670402916959761216, −5.28675861042101974648083818726, −4.72675008489471344753420479373, −4.46812707697114218135147284080, −4.01284167610470477709586536305, −3.48484597716519707112849320087, −3.42608763303116007804076750284, −2.49754697930114453671279577753, −2.41799967417037184539338661146, −1.93803534830207011141089981592, −1.85707531126915089180039643168, −1.10515372051200958988874129191, −0.43723260840564852513525869336,
0.43723260840564852513525869336, 1.10515372051200958988874129191, 1.85707531126915089180039643168, 1.93803534830207011141089981592, 2.41799967417037184539338661146, 2.49754697930114453671279577753, 3.42608763303116007804076750284, 3.48484597716519707112849320087, 4.01284167610470477709586536305, 4.46812707697114218135147284080, 4.72675008489471344753420479373, 5.28675861042101974648083818726, 5.48401149006670402916959761216, 6.04791937644171736171060806038, 6.25703806751738242539654939238, 6.47003123866795944021666941504, 6.65554246258793523144270584452, 7.07891161020720600668814371528, 7.79221960270476948666552507158, 7.913769290928375486450121330831
