| L(s) = 1 | + 2-s + 4-s + 6·7-s + 8-s + 6·14-s + 16-s + 6·19-s + 8·25-s + 6·28-s + 2·29-s + 32-s + 6·38-s − 4·41-s + 14·49-s + 8·50-s − 4·53-s + 6·56-s + 2·58-s − 22·59-s + 6·61-s + 64-s − 14·71-s − 8·73-s + 6·76-s − 4·82-s + 18·89-s + 14·98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.26·7-s + 0.353·8-s + 1.60·14-s + 1/4·16-s + 1.37·19-s + 8/5·25-s + 1.13·28-s + 0.371·29-s + 0.176·32-s + 0.973·38-s − 0.624·41-s + 2·49-s + 1.13·50-s − 0.549·53-s + 0.801·56-s + 0.262·58-s − 2.86·59-s + 0.768·61-s + 1/8·64-s − 1.66·71-s − 0.936·73-s + 0.688·76-s − 0.441·82-s + 1.90·89-s + 1.41·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.483536939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.483536939\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.581530589404496518815842658483, −7.932475408683680978145000161438, −7.58490375611932214809343618425, −7.39092763053796333155157617656, −6.69083736281150804763253032228, −6.15182652242368372528476516781, −5.63895139976281798185269800775, −4.97972265100455512669113024170, −4.78416482939621821396337662767, −4.58924705614683252130519851788, −3.64065026851299280748401695418, −3.11949215010368454758921333057, −2.46237859507183005211996522564, −1.58085336106198186873536705957, −1.21876207985736478325886318005,
1.21876207985736478325886318005, 1.58085336106198186873536705957, 2.46237859507183005211996522564, 3.11949215010368454758921333057, 3.64065026851299280748401695418, 4.58924705614683252130519851788, 4.78416482939621821396337662767, 4.97972265100455512669113024170, 5.63895139976281798185269800775, 6.15182652242368372528476516781, 6.69083736281150804763253032228, 7.39092763053796333155157617656, 7.58490375611932214809343618425, 7.932475408683680978145000161438, 8.581530589404496518815842658483
