| L(s) = 1 | + 2·3-s + 3·9-s + 6·13-s − 4·17-s − 8·23-s + 6·25-s + 4·27-s + 20·29-s + 12·39-s + 8·43-s + 10·49-s − 8·51-s + 12·53-s − 4·61-s − 16·69-s + 12·75-s + 5·81-s + 40·87-s − 4·101-s + 32·103-s + 16·107-s + 28·113-s + 18·117-s + 22·121-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.66·13-s − 0.970·17-s − 1.66·23-s + 6/5·25-s + 0.769·27-s + 3.71·29-s + 1.92·39-s + 1.21·43-s + 10/7·49-s − 1.12·51-s + 1.64·53-s − 0.512·61-s − 1.92·69-s + 1.38·75-s + 5/9·81-s + 4.28·87-s − 0.398·101-s + 3.15·103-s + 1.54·107-s + 2.63·113-s + 1.66·117-s + 2·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.423517985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.423517985\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.979849600839960952138747224672, −8.669029149408725192209302403534, −8.386791930404909916333833681284, −8.291205513783373362182260442060, −7.49330045732221864983296555148, −7.41278339185121496125920940011, −6.77894512551745001712123132310, −6.42263811701478597611579039304, −6.10915943981464401815743566984, −5.82750814960662168044895062646, −4.98453424994690680497479421768, −4.62503532348734583399201605139, −4.27421890699668819415846303902, −3.86112763593529865221040581868, −3.39332894661611031744321486052, −2.91639152398017264802031822777, −2.36411447835888112330154150030, −2.13213419224769988985383921591, −1.12805404260112906456073730735, −0.865190569043465432791976012180,
0.865190569043465432791976012180, 1.12805404260112906456073730735, 2.13213419224769988985383921591, 2.36411447835888112330154150030, 2.91639152398017264802031822777, 3.39332894661611031744321486052, 3.86112763593529865221040581868, 4.27421890699668819415846303902, 4.62503532348734583399201605139, 4.98453424994690680497479421768, 5.82750814960662168044895062646, 6.10915943981464401815743566984, 6.42263811701478597611579039304, 6.77894512551745001712123132310, 7.41278339185121496125920940011, 7.49330045732221864983296555148, 8.291205513783373362182260442060, 8.386791930404909916333833681284, 8.669029149408725192209302403534, 8.979849600839960952138747224672
