| L(s) = 1 | − 2·5-s + 4·7-s − 2·13-s + 2·19-s + 3·25-s + 4·31-s − 8·35-s + 10·37-s + 12·41-s + 4·43-s − 12·47-s − 2·49-s + 18·53-s + 4·61-s + 4·65-s + 10·67-s + 12·71-s − 8·73-s − 8·79-s + 24·89-s − 8·91-s − 4·95-s − 2·97-s + 12·101-s + 10·103-s − 6·107-s − 8·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.458·19-s + 3/5·25-s + 0.718·31-s − 1.35·35-s + 1.64·37-s + 1.87·41-s + 0.609·43-s − 1.75·47-s − 2/7·49-s + 2.47·53-s + 0.512·61-s + 0.496·65-s + 1.22·67-s + 1.42·71-s − 0.936·73-s − 0.900·79-s + 2.54·89-s − 0.838·91-s − 0.410·95-s − 0.203·97-s + 1.19·101-s + 0.985·103-s − 0.580·107-s − 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.286879226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.286879226\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.500944837848202769823797012696, −8.419560780479533498674393881632, −7.927790483488419868314294261614, −7.84374712086266629852550336834, −7.29615915720755471268165866780, −7.22070268779421723856344852706, −6.45814076891746369125045542479, −6.33362801843414492216741919866, −5.54370577975522310387811567902, −5.44835884911990703875667087530, −4.78589756435213296197656594597, −4.66121459892390849198714665659, −4.13223568276784652380286276395, −3.95291295420736884426522019352, −3.12292382116067265081520818531, −2.91433520063197880129429776464, −2.12514771523704811962369045359, −1.93274918210383358514914734067, −0.880701398752982840553281107725, −0.75052375508716893642255794850,
0.75052375508716893642255794850, 0.880701398752982840553281107725, 1.93274918210383358514914734067, 2.12514771523704811962369045359, 2.91433520063197880129429776464, 3.12292382116067265081520818531, 3.95291295420736884426522019352, 4.13223568276784652380286276395, 4.66121459892390849198714665659, 4.78589756435213296197656594597, 5.44835884911990703875667087530, 5.54370577975522310387811567902, 6.33362801843414492216741919866, 6.45814076891746369125045542479, 7.22070268779421723856344852706, 7.29615915720755471268165866780, 7.84374712086266629852550336834, 7.927790483488419868314294261614, 8.419560780479533498674393881632, 8.500944837848202769823797012696
