| L(s) = 1 | + 5·9-s + 12·11-s + 16·19-s − 6·29-s − 4·31-s + 6·41-s + 2·61-s + 8·79-s + 16·81-s − 6·89-s + 60·99-s + 6·101-s − 34·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 80·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 3.61·11-s + 3.67·19-s − 1.11·29-s − 0.718·31-s + 0.937·41-s + 0.256·61-s + 0.900·79-s + 16/9·81-s − 0.635·89-s + 6.03·99-s + 0.597·101-s − 3.25·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 6.11·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.714372756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.714372756\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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
−8.515325812594922170917733545634, −7.86752332192373604831485677850, −7.64454205141315531833549156978, −7.28891948042993951860601591798, −6.96732521651290807865312985415, −6.81313793144135731210537748877, −6.37125555402403764607217372538, −5.97476631605938133342484026571, −5.39342702097849832509974697530, −5.33080307114743325718367476744, −4.56312248845335766050668530885, −4.33132987979400461158742265795, −3.82722816236990703944666985505, −3.60589671892736616441633664894, −3.42093321071377068114770700622, −2.69657897532409626969289104444, −1.86702851527899478090995427122, −1.46634521361575434593484116835, −1.16852305385555366048474044306, −0.870041381074443320868050133853,
0.870041381074443320868050133853, 1.16852305385555366048474044306, 1.46634521361575434593484116835, 1.86702851527899478090995427122, 2.69657897532409626969289104444, 3.42093321071377068114770700622, 3.60589671892736616441633664894, 3.82722816236990703944666985505, 4.33132987979400461158742265795, 4.56312248845335766050668530885, 5.33080307114743325718367476744, 5.39342702097849832509974697530, 5.97476631605938133342484026571, 6.37125555402403764607217372538, 6.81313793144135731210537748877, 6.96732521651290807865312985415, 7.28891948042993951860601591798, 7.64454205141315531833549156978, 7.86752332192373604831485677850, 8.515325812594922170917733545634
