| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 9-s + 2·14-s − 16-s − 12·17-s − 18-s − 6·25-s + 2·28-s − 5·32-s + 12·34-s − 36-s + 4·41-s + 3·49-s + 6·50-s − 6·56-s − 2·63-s + 7·64-s + 12·68-s + 3·72-s − 12·73-s − 32·79-s + 81-s − 4·82-s − 28·89-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.534·14-s − 1/4·16-s − 2.91·17-s − 0.235·18-s − 6/5·25-s + 0.377·28-s − 0.883·32-s + 2.05·34-s − 1/6·36-s + 0.624·41-s + 3/7·49-s + 0.848·50-s − 0.801·56-s − 0.251·63-s + 7/8·64-s + 1.45·68-s + 0.353·72-s − 1.40·73-s − 3.60·79-s + 1/9·81-s − 0.441·82-s − 2.96·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−10.20179794179912608851597548824, −9.724661210726173973340366981146, −9.238117136499894867345980510658, −8.672365196683144779961098572501, −8.535675361637489516400731752225, −7.48171223884197946068743795147, −7.25047783802838427330028450541, −6.47604188152629865155494776849, −6.00140300914464627775014931904, −5.08548839023881336508605977410, −4.19315915887555872024919481174, −4.13559084050773741974089362056, −2.78781035361267366934893473837, −1.80984559120091606425042866771, 0,
1.80984559120091606425042866771, 2.78781035361267366934893473837, 4.13559084050773741974089362056, 4.19315915887555872024919481174, 5.08548839023881336508605977410, 6.00140300914464627775014931904, 6.47604188152629865155494776849, 7.25047783802838427330028450541, 7.48171223884197946068743795147, 8.535675361637489516400731752225, 8.672365196683144779961098572501, 9.238117136499894867345980510658, 9.724661210726173973340366981146, 10.20179794179912608851597548824
