L(s) = 1 | + 4·3-s − 3·4-s + 2·5-s + 6·9-s − 12·12-s + 8·15-s + 5·16-s − 6·20-s + 4·23-s − 7·25-s − 4·27-s − 4·31-s − 18·36-s − 6·37-s + 12·45-s + 4·47-s + 20·48-s − 10·49-s + 18·53-s + 16·59-s − 24·60-s − 3·64-s + 4·67-s + 16·69-s + 24·71-s − 28·75-s + 10·80-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 3/2·4-s + 0.894·5-s + 2·9-s − 3.46·12-s + 2.06·15-s + 5/4·16-s − 1.34·20-s + 0.834·23-s − 7/5·25-s − 0.769·27-s − 0.718·31-s − 3·36-s − 0.986·37-s + 1.78·45-s + 0.583·47-s + 2.88·48-s − 1.42·49-s + 2.47·53-s + 2.08·59-s − 3.09·60-s − 3/8·64-s + 0.488·67-s + 1.92·69-s + 2.84·71-s − 3.23·75-s + 1.11·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.699138266\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699138266\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 11 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10272305542928838383025084052, −9.975450561586109317601560382100, −9.950218896470013175355990146225, −9.182457967238804819066801847611, −9.127942331541421284163370204249, −8.356936232923193217562617522792, −8.264856530445176716222542589828, −7.49323810887805876503518242868, −6.72221402891887535126565651102, −5.44861874763839175210229504778, −5.38415572352897256696127903868, −3.89751875979123562820900427499, −3.84177902362047353880486406829, −2.75274084463573300677827364197, −1.98218366620434510274683932709,
1.98218366620434510274683932709, 2.75274084463573300677827364197, 3.84177902362047353880486406829, 3.89751875979123562820900427499, 5.38415572352897256696127903868, 5.44861874763839175210229504778, 6.72221402891887535126565651102, 7.49323810887805876503518242868, 8.264856530445176716222542589828, 8.356936232923193217562617522792, 9.127942331541421284163370204249, 9.182457967238804819066801847611, 9.950218896470013175355990146225, 9.975450561586109317601560382100, 11.10272305542928838383025084052