| L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s + 16-s − 6·17-s − 2·19-s − 6·22-s − 10·25-s + 32-s − 6·34-s − 2·38-s + 18·41-s − 2·43-s − 6·44-s − 10·49-s − 10·50-s + 6·59-s + 64-s + 10·67-s − 6·68-s + 22·73-s − 2·76-s + 18·82-s + 24·83-s − 2·86-s − 6·88-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 1.27·22-s − 2·25-s + 0.176·32-s − 1.02·34-s − 0.324·38-s + 2.81·41-s − 0.304·43-s − 0.904·44-s − 1.42·49-s − 1.41·50-s + 0.781·59-s + 1/8·64-s + 1.22·67-s − 0.727·68-s + 2.57·73-s − 0.229·76-s + 1.98·82-s + 2.63·83-s − 0.215·86-s − 0.639·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.925402993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925402993\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.004874510471126629875326215800, −7.63671497466111343831458484773, −7.61550392042635002094679885373, −6.71782060930200301497522083770, −6.25936593777468405983226275094, −6.13537847248252222252378323029, −5.27398052675085301748339586603, −5.13805326110939411578475614907, −4.58847837490118018713372112605, −3.94847330019587176084970982031, −3.66068742781442559176084506010, −2.78620746794391482576765657021, −2.19879385611801234222671009191, −2.08674986394077854980553765067, −0.55030123589812757559608034041,
0.55030123589812757559608034041, 2.08674986394077854980553765067, 2.19879385611801234222671009191, 2.78620746794391482576765657021, 3.66068742781442559176084506010, 3.94847330019587176084970982031, 4.58847837490118018713372112605, 5.13805326110939411578475614907, 5.27398052675085301748339586603, 6.13537847248252222252378323029, 6.25936593777468405983226275094, 6.71782060930200301497522083770, 7.61550392042635002094679885373, 7.63671497466111343831458484773, 8.004874510471126629875326215800
