| L(s) = 1 | − 4·3-s + 58·9-s + 64·16-s − 26·17-s + 156·23-s − 78·25-s − 416·27-s − 394·29-s − 312·43-s − 256·48-s − 286·49-s + 104·51-s + 372·53-s − 290·61-s − 624·69-s + 312·75-s − 304·79-s + 1.96e3·81-s + 1.57e3·87-s − 1.63e3·101-s − 6.55e3·103-s − 1.04e3·107-s − 654·113-s − 1.63e3·121-s + 127-s + 1.24e3·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 2.14·9-s + 16-s − 0.370·17-s + 1.41·23-s − 0.623·25-s − 2.96·27-s − 2.52·29-s − 1.10·43-s − 0.769·48-s − 0.833·49-s + 0.285·51-s + 0.964·53-s − 0.608·61-s − 1.08·69-s + 0.480·75-s − 0.432·79-s + 2.68·81-s + 1.94·87-s − 1.61·101-s − 6.26·103-s − 0.943·107-s − 0.544·113-s − 1.23·121-s + 0.000698·127-s + 0.851·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2694747472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2694747472\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} )( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} ) \) |
| 3 | $C_2^2$ | \( ( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 39 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 397 T^{2} + p^{6} T^{4} )( 1 + 683 T^{2} + p^{6} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 1638 T^{2} + 911483 T^{4} + 1638 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 13 T - 4744 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 12818 T^{2} + 117255243 T^{4} + 12818 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 197 T + 14420 T^{2} + 197 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 54106 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 49777 T^{2} - 87976680 T^{4} + 49777 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 110617 T^{2} + 7486016448 T^{4} + 110617 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 156 T - 55171 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 181402 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 335738 T^{2} + 70539471003 T^{4} - 335738 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 141518 T^{2} - 70431037845 T^{4} - 141518 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 288106 T^{2} - 45095216685 T^{4} + 288106 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 731809 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 749190 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 1339182 T^{2} + 1296427138163 T^{4} + 1339182 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 1768702 T^{2} + 2295334759875 T^{4} + 1768702 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.147633935025717020481531939898, −8.726663677637035073899286257173, −8.016155787730277047372686460105, −7.933881390920548303463260050649, −7.84204876759055956765639938549, −7.65691460233254945002174343382, −6.93545204156817504678880127090, −6.83875942143023981378150222726, −6.80222687969741269464423094826, −6.60916279389742401079830992922, −5.79331407752267733878280277254, −5.56497464752801068374341290045, −5.42685636239814820828882099831, −5.40454543706290499328685267004, −4.71314616758000778788612895226, −4.23142765820583406878319203939, −4.11360195887226238013533721709, −3.96009558417502354075370650205, −3.23488663651536556725432799365, −3.14978761307355954195363194354, −2.32061267747781760103864705700, −1.85042139776134295975923511880, −1.37234244543934425469486742792, −1.21806263970241569955795356761, −0.11977577245846564232444661477,
0.11977577245846564232444661477, 1.21806263970241569955795356761, 1.37234244543934425469486742792, 1.85042139776134295975923511880, 2.32061267747781760103864705700, 3.14978761307355954195363194354, 3.23488663651536556725432799365, 3.96009558417502354075370650205, 4.11360195887226238013533721709, 4.23142765820583406878319203939, 4.71314616758000778788612895226, 5.40454543706290499328685267004, 5.42685636239814820828882099831, 5.56497464752801068374341290045, 5.79331407752267733878280277254, 6.60916279389742401079830992922, 6.80222687969741269464423094826, 6.83875942143023981378150222726, 6.93545204156817504678880127090, 7.65691460233254945002174343382, 7.84204876759055956765639938549, 7.933881390920548303463260050649, 8.016155787730277047372686460105, 8.726663677637035073899286257173, 9.147633935025717020481531939898
