| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 2·7-s − 4·8-s + 4·10-s − 4·11-s + 4·14-s + 5·16-s − 4·17-s + 10·19-s − 6·20-s + 8·22-s + 2·23-s − 25-s − 6·28-s + 4·29-s + 12·31-s − 6·32-s + 8·34-s + 4·35-s + 4·37-s − 20·38-s + 8·40-s + 4·43-s − 12·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s − 1.41·8-s + 1.26·10-s − 1.20·11-s + 1.06·14-s + 5/4·16-s − 0.970·17-s + 2.29·19-s − 1.34·20-s + 1.70·22-s + 0.417·23-s − 1/5·25-s − 1.13·28-s + 0.742·29-s + 2.15·31-s − 1.06·32-s + 1.37·34-s + 0.676·35-s + 0.657·37-s − 3.24·38-s + 1.26·40-s + 0.609·43-s − 1.80·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7250156492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7250156492\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.962461978981467558591432126627, −9.779248438971436790349631236950, −8.981296378570230157710136272814, −8.947689521205447726793047647873, −8.359718194991733915568186002825, −8.050057113882701124054777004681, −7.43672336412669851928003288646, −7.40958691694476786450167906856, −6.98670064856040608119055429591, −6.36358796041517461159700235275, −5.93193861521945797418417525525, −5.54119488052097393706486540617, −4.68782173782749819707546711224, −4.57458447192399199202565340230, −3.44932887505211656120201533414, −3.34750263310615622435749510052, −2.53783335702546801626679811217, −2.33543782304875522931779876442, −0.992816037072599654824390022916, −0.60053761833786435033192798039,
0.60053761833786435033192798039, 0.992816037072599654824390022916, 2.33543782304875522931779876442, 2.53783335702546801626679811217, 3.34750263310615622435749510052, 3.44932887505211656120201533414, 4.57458447192399199202565340230, 4.68782173782749819707546711224, 5.54119488052097393706486540617, 5.93193861521945797418417525525, 6.36358796041517461159700235275, 6.98670064856040608119055429591, 7.40958691694476786450167906856, 7.43672336412669851928003288646, 8.050057113882701124054777004681, 8.359718194991733915568186002825, 8.947689521205447726793047647873, 8.981296378570230157710136272814, 9.779248438971436790349631236950, 9.962461978981467558591432126627
