| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·9-s − 2·10-s − 2·13-s − 16-s + 10·17-s + 2·18-s − 2·20-s − 7·25-s + 2·26-s − 18·29-s − 5·32-s − 10·34-s + 2·36-s − 6·37-s + 6·40-s + 10·41-s − 4·45-s − 10·49-s + 7·50-s + 2·52-s + 18·53-s + 18·58-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 2/3·9-s − 0.632·10-s − 0.554·13-s − 1/4·16-s + 2.42·17-s + 0.471·18-s − 0.447·20-s − 7/5·25-s + 0.392·26-s − 3.34·29-s − 0.883·32-s − 1.71·34-s + 1/3·36-s − 0.986·37-s + 0.948·40-s + 1.56·41-s − 0.596·45-s − 1.42·49-s + 0.989·50-s + 0.277·52-s + 2.47·53-s + 2.36·58-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.043042867788870716539415194167, −8.264856530445176716222542589828, −7.78091021638599658560987714578, −7.49323810887805876503518242868, −7.18510639953264999652502212264, −6.10386760784300470004944448119, −5.73468060896468798044264674509, −5.44861874763839175210229504778, −5.04523846649891070859056262216, −3.89751875979123562820900427499, −3.79587121545739932046191560595, −2.85249788770459737210058313708, −1.98218366620434510274683932709, −1.38254023963820058194153479990, 0,
1.38254023963820058194153479990, 1.98218366620434510274683932709, 2.85249788770459737210058313708, 3.79587121545739932046191560595, 3.89751875979123562820900427499, 5.04523846649891070859056262216, 5.44861874763839175210229504778, 5.73468060896468798044264674509, 6.10386760784300470004944448119, 7.18510639953264999652502212264, 7.49323810887805876503518242868, 7.78091021638599658560987714578, 8.264856530445176716222542589828, 9.043042867788870716539415194167
