| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s − 2·9-s − 2·14-s − 16-s + 8·17-s + 2·18-s − 18·23-s − 25-s − 2·28-s − 5·32-s − 8·34-s + 2·36-s + 10·41-s + 18·46-s + 8·47-s − 11·49-s + 50-s + 6·56-s − 4·63-s + 7·64-s − 8·68-s − 16·71-s − 6·72-s − 22·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 2/3·9-s − 0.534·14-s − 1/4·16-s + 1.94·17-s + 0.471·18-s − 3.75·23-s − 1/5·25-s − 0.377·28-s − 0.883·32-s − 1.37·34-s + 1/3·36-s + 1.56·41-s + 2.65·46-s + 1.16·47-s − 1.57·49-s + 0.141·50-s + 0.801·56-s − 0.503·63-s + 7/8·64-s − 0.970·68-s − 1.89·71-s − 0.707·72-s − 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.569165758261804870177868073898, −8.287570693733701613432166498970, −7.85884968360616883145053235731, −7.66039763333771123227596439320, −7.12251875143434015278201977633, −6.08852912831720735332378674864, −5.72640884385550487126915198965, −5.60157612088078313975836309422, −4.59002554505366699590804892989, −4.25666915587159511841100490901, −3.69588602176779207669631056609, −2.89089326170447943989229722201, −1.96512649323539838938700359285, −1.31708422340871154028668296964, 0,
1.31708422340871154028668296964, 1.96512649323539838938700359285, 2.89089326170447943989229722201, 3.69588602176779207669631056609, 4.25666915587159511841100490901, 4.59002554505366699590804892989, 5.60157612088078313975836309422, 5.72640884385550487126915198965, 6.08852912831720735332378674864, 7.12251875143434015278201977633, 7.66039763333771123227596439320, 7.85884968360616883145053235731, 8.287570693733701613432166498970, 8.569165758261804870177868073898
