L(s) = 1 | − 4-s + 4·7-s − 6·9-s − 8·11-s + 16-s − 4·28-s + 6·36-s + 12·37-s − 4·41-s + 8·44-s − 12·47-s − 2·49-s − 24·53-s − 24·63-s − 64-s − 32·67-s − 24·71-s − 12·73-s − 32·77-s + 27·81-s + 24·83-s + 48·99-s − 12·101-s + 4·112-s + 26·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.51·7-s − 2·9-s − 2.41·11-s + 1/4·16-s − 0.755·28-s + 36-s + 1.97·37-s − 0.624·41-s + 1.20·44-s − 1.75·47-s − 2/7·49-s − 3.29·53-s − 3.02·63-s − 1/8·64-s − 3.90·67-s − 2.84·71-s − 1.40·73-s − 3.64·77-s + 3·81-s + 2.63·83-s + 4.82·99-s − 1.19·101-s + 0.377·112-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09336431676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09336431676\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.529476983241532627680476870291, −8.732261922978906109452544611839, −8.708590372648492271952893423697, −8.182993013279282741106026055970, −7.896338251726870685201737463629, −7.69001581325046600009766621716, −7.47870637464258155053450181997, −6.47277172754340087586948320194, −5.97181277637094020353152813282, −5.94440288677022070981515112631, −5.24705458824910786220767301183, −4.96674096373782225965926081953, −4.69229767858941919188746453026, −4.36878112753486787173983706564, −3.22449225766080697939821768864, −3.04931326047368461988060363974, −2.74013117932785747176195496757, −1.95647642534709988821008627475, −1.42822410044774523385856348133, −0.10972886878317483485932969907,
0.10972886878317483485932969907, 1.42822410044774523385856348133, 1.95647642534709988821008627475, 2.74013117932785747176195496757, 3.04931326047368461988060363974, 3.22449225766080697939821768864, 4.36878112753486787173983706564, 4.69229767858941919188746453026, 4.96674096373782225965926081953, 5.24705458824910786220767301183, 5.94440288677022070981515112631, 5.97181277637094020353152813282, 6.47277172754340087586948320194, 7.47870637464258155053450181997, 7.69001581325046600009766621716, 7.896338251726870685201737463629, 8.182993013279282741106026055970, 8.708590372648492271952893423697, 8.732261922978906109452544611839, 9.529476983241532627680476870291