| L(s) = 1 | − 6·3-s − 8·4-s + 27·9-s + 48·12-s + 48·16-s − 50·25-s − 108·27-s − 216·36-s − 166·43-s − 288·48-s − 71·49-s + 94·61-s − 256·64-s + 300·75-s + 22·79-s + 405·81-s + 400·100-s + 74·103-s + 864·108-s − 242·121-s + 127-s + 996·129-s + 131-s + 137-s + 139-s + 1.29e3·144-s + 426·147-s + ⋯ |
| L(s) = 1 | − 2·3-s − 2·4-s + 3·9-s + 4·12-s + 3·16-s − 2·25-s − 4·27-s − 6·36-s − 3.86·43-s − 6·48-s − 1.44·49-s + 1.54·61-s − 4·64-s + 4·75-s + 0.278·79-s + 5·81-s + 4·100-s + 0.718·103-s + 8·108-s − 2·121-s + 0.00787·127-s + 7.72·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 9·144-s + 2.89·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02491921325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02491921325\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 71 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1753 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 83 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8809 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9791 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9743 T^{2} + p^{4} T^{4} \) |
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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
−11.44991479999199356076198421935, −10.16961375600119114479483554592, −10.15793858406704662144799861961, −9.638072357310089696671655486896, −9.549761287344916474844607912477, −8.635355498690974045557840293225, −8.297864957653650096361961672631, −7.78346035791073796998436889994, −7.29552220352684401657357589807, −6.56630307761227212892235164188, −6.20637050447149906018165278296, −5.64063605092396910060379066126, −5.20830954624014977811487832864, −4.86723775910471708259392626933, −4.47576016634325963831985491339, −3.64885451617794452956861625395, −3.61728499200282887731208212879, −1.84085008093400730460935657014, −1.14287225696603410811504424287, −0.087720024789181745540025398670,
0.087720024789181745540025398670, 1.14287225696603410811504424287, 1.84085008093400730460935657014, 3.61728499200282887731208212879, 3.64885451617794452956861625395, 4.47576016634325963831985491339, 4.86723775910471708259392626933, 5.20830954624014977811487832864, 5.64063605092396910060379066126, 6.20637050447149906018165278296, 6.56630307761227212892235164188, 7.29552220352684401657357589807, 7.78346035791073796998436889994, 8.297864957653650096361961672631, 8.635355498690974045557840293225, 9.549761287344916474844607912477, 9.638072357310089696671655486896, 10.15793858406704662144799861961, 10.16961375600119114479483554592, 11.44991479999199356076198421935
