| L(s) = 1 | + 2·3-s − 4-s − 2·8-s + 4·11-s − 2·12-s + 3·13-s + 16-s + 2·17-s + 13·19-s + 5·23-s − 4·24-s − 4·25-s − 2·27-s − 29-s − 2·31-s + 4·32-s + 8·33-s + 2·37-s + 6·39-s − 6·41-s + 2·43-s − 4·44-s + 2·48-s + 2·49-s + 4·51-s − 3·52-s − 16·53-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.707·8-s + 1.20·11-s − 0.577·12-s + 0.832·13-s + 1/4·16-s + 0.485·17-s + 2.98·19-s + 1.04·23-s − 0.816·24-s − 4/5·25-s − 0.384·27-s − 0.185·29-s − 0.359·31-s + 0.707·32-s + 1.39·33-s + 0.328·37-s + 0.960·39-s − 0.937·41-s + 0.304·43-s − 0.603·44-s + 0.288·48-s + 2/7·49-s + 0.560·51-s − 0.416·52-s − 2.19·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340114 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340114 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.703919325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.703919325\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 170057 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 182 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 42 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 48 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 48 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−13.0953743341, −12.6838802026, −11.9424074411, −11.7898263651, −11.5010065183, −11.0401752268, −10.2447929472, −9.88006376669, −9.38005135462, −9.13965205191, −8.93549831423, −8.34860423895, −7.94437786117, −7.46808448772, −7.04963957813, −6.35357214026, −5.89033557254, −5.41266168343, −4.92518537822, −4.10119469824, −3.54740244821, −3.13562255571, −2.91094515111, −1.65716898457, −0.981917411337,
0.981917411337, 1.65716898457, 2.91094515111, 3.13562255571, 3.54740244821, 4.10119469824, 4.92518537822, 5.41266168343, 5.89033557254, 6.35357214026, 7.04963957813, 7.46808448772, 7.94437786117, 8.34860423895, 8.93549831423, 9.13965205191, 9.38005135462, 9.88006376669, 10.2447929472, 11.0401752268, 11.5010065183, 11.7898263651, 11.9424074411, 12.6838802026, 13.0953743341
