| L(s) = 1 | + 2-s + 4-s − 2·7-s + 3·8-s + 2·11-s + 5·13-s − 2·14-s + 16-s + 7·17-s + 6·19-s + 2·22-s + 9·23-s + 5·26-s − 2·28-s − 2·29-s − 3·31-s − 32-s + 7·34-s + 37-s + 6·38-s + 3·41-s + 3·43-s + 2·44-s + 9·46-s + 15·47-s − 11·49-s + 5·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 1.06·8-s + 0.603·11-s + 1.38·13-s − 0.534·14-s + 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.426·22-s + 1.87·23-s + 0.980·26-s − 0.377·28-s − 0.371·29-s − 0.538·31-s − 0.176·32-s + 1.20·34-s + 0.164·37-s + 0.973·38-s + 0.468·41-s + 0.457·43-s + 0.301·44-s + 1.32·46-s + 2.18·47-s − 1.57·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.833669291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.833669291\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_4$ | \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 210 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 - 26 T + 330 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 181 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.027994156606032544069874377355, −9.008353773601482822666384632232, −8.317427360164500719333675615895, −7.73340111824441623790719979833, −7.53549529188499791817938419355, −7.18263939899137526638170993091, −6.91381696389274490866972809032, −6.26510694885346203851524678458, −5.89337302097861147499773196989, −5.79500240746988737288128213927, −5.12127610624148283958922911689, −4.86949119882560381764474549798, −4.21525007864685896073238498892, −3.91690925270182544016244310188, −3.25117196511049350532660073447, −3.21110396665105094870106843254, −2.70762100408032687538616862079, −1.68082303157079390627616932394, −1.33334055688953324011476936597, −0.810047830414422836938202944158,
0.810047830414422836938202944158, 1.33334055688953324011476936597, 1.68082303157079390627616932394, 2.70762100408032687538616862079, 3.21110396665105094870106843254, 3.25117196511049350532660073447, 3.91690925270182544016244310188, 4.21525007864685896073238498892, 4.86949119882560381764474549798, 5.12127610624148283958922911689, 5.79500240746988737288128213927, 5.89337302097861147499773196989, 6.26510694885346203851524678458, 6.91381696389274490866972809032, 7.18263939899137526638170993091, 7.53549529188499791817938419355, 7.73340111824441623790719979833, 8.317427360164500719333675615895, 9.008353773601482822666384632232, 9.027994156606032544069874377355
