| L(s) = 1 | − 6·3-s + 4-s − 2·5-s + 21·9-s − 2·11-s − 6·12-s + 12·15-s + 16-s − 2·20-s − 8·23-s − 7·25-s − 54·27-s + 8·31-s + 12·33-s + 21·36-s + 6·37-s − 2·44-s − 42·45-s + 26·47-s − 6·48-s − 13·49-s + 24·53-s + 4·55-s − 20·59-s + 12·60-s + 64-s − 4·67-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 1/2·4-s − 0.894·5-s + 7·9-s − 0.603·11-s − 1.73·12-s + 3.09·15-s + 1/4·16-s − 0.447·20-s − 1.66·23-s − 7/5·25-s − 10.3·27-s + 1.43·31-s + 2.08·33-s + 7/2·36-s + 0.986·37-s − 0.301·44-s − 6.26·45-s + 3.79·47-s − 0.866·48-s − 1.85·49-s + 3.29·53-s + 0.539·55-s − 2.60·59-s + 1.54·60-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81796 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81796 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.982600302630202160398757175989, −9.182231187189023046155985298119, −8.174465989121661359437511477893, −7.52260855659566213355764232106, −7.45985560667588965978165906144, −6.69700065044511212992831284617, −6.06963680351175114271015390023, −5.88908450915949561478136129218, −5.58007316868712318029783390276, −4.61153453491182141362175201622, −4.43660190737283567298107688401, −3.76677853005733776527358303686, −2.25144343695204805750190070200, −0.973879971324260460585952773727, 0,
0.973879971324260460585952773727, 2.25144343695204805750190070200, 3.76677853005733776527358303686, 4.43660190737283567298107688401, 4.61153453491182141362175201622, 5.58007316868712318029783390276, 5.88908450915949561478136129218, 6.06963680351175114271015390023, 6.69700065044511212992831284617, 7.45985560667588965978165906144, 7.52260855659566213355764232106, 8.174465989121661359437511477893, 9.182231187189023046155985298119, 9.982600302630202160398757175989
