| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s − 2·9-s + 12-s + 2·14-s + 16-s + 2·18-s + 2·19-s − 2·21-s − 24-s + 6·25-s − 5·27-s − 2·28-s − 2·29-s − 32-s − 2·36-s − 2·38-s − 2·41-s + 2·42-s + 6·43-s + 48-s + 5·49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.458·19-s − 0.436·21-s − 0.204·24-s + 6/5·25-s − 0.962·27-s − 0.377·28-s − 0.371·29-s − 0.176·32-s − 1/3·36-s − 0.324·38-s − 0.312·41-s + 0.308·42-s + 0.914·43-s + 0.144·48-s + 5/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208800897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208800897\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.259712654241396796547311771474, −7.82605340888945997265470134743, −7.46395482220022039924279115142, −6.93116987999752998749237098000, −6.61086360859203377855533629462, −6.05289959677352369003533591547, −5.54065332434140585615740783291, −5.25249119343827918777312127648, −4.38574167408013912303336462185, −3.88634169470465792105439982583, −3.23213552570253269079230658771, −2.85762836432667921636600379811, −2.38656568360663883466613103177, −1.53089148414040448160427671310, −0.57158065856009092650012287122,
0.57158065856009092650012287122, 1.53089148414040448160427671310, 2.38656568360663883466613103177, 2.85762836432667921636600379811, 3.23213552570253269079230658771, 3.88634169470465792105439982583, 4.38574167408013912303336462185, 5.25249119343827918777312127648, 5.54065332434140585615740783291, 6.05289959677352369003533591547, 6.61086360859203377855533629462, 6.93116987999752998749237098000, 7.46395482220022039924279115142, 7.82605340888945997265470134743, 8.259712654241396796547311771474
