| L(s) = 1 | + 4·3-s + 6·9-s + 4·11-s + 2·13-s + 6·17-s − 8·19-s + 10·25-s − 4·27-s + 16·33-s + 2·37-s + 8·39-s + 20·47-s − 5·49-s + 24·51-s − 32·57-s − 16·67-s − 4·71-s + 40·75-s − 37·81-s + 14·89-s + 24·99-s − 6·101-s − 30·109-s + 8·111-s − 2·113-s + 12·117-s − 10·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s + 2·25-s − 0.769·27-s + 2.78·33-s + 0.328·37-s + 1.28·39-s + 2.91·47-s − 5/7·49-s + 3.36·51-s − 4.23·57-s − 1.95·67-s − 0.474·71-s + 4.61·75-s − 4.11·81-s + 1.48·89-s + 2.41·99-s − 0.597·101-s − 2.87·109-s + 0.759·111-s − 0.188·113-s + 1.10·117-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.657897293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.657897293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 157 | $C_2$ | \( 1 + 18 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 99 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
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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
−10.62504155870259967431267237679, −10.45778215425951597705850617163, −9.564040056767187668964397568384, −9.452820608061107156971105416785, −8.878467033920049179090364089141, −8.680995183118574253524087588320, −8.421917864294978354980839239599, −7.929891391194526027977309513048, −7.34871972972432293749043036046, −7.12383245070013416844393322299, −6.20633611310095530759514643449, −6.12556684732521930357550277691, −5.31258618107902629744169190432, −4.53438804339524701530594907660, −3.83611110176451544588490335940, −3.81286337712981380355021196310, −2.86263977243619796173848743339, −2.78541376566271024081592934868, −1.90696206171907722532035588325, −1.17511020827933720759707733558,
1.17511020827933720759707733558, 1.90696206171907722532035588325, 2.78541376566271024081592934868, 2.86263977243619796173848743339, 3.81286337712981380355021196310, 3.83611110176451544588490335940, 4.53438804339524701530594907660, 5.31258618107902629744169190432, 6.12556684732521930357550277691, 6.20633611310095530759514643449, 7.12383245070013416844393322299, 7.34871972972432293749043036046, 7.929891391194526027977309513048, 8.421917864294978354980839239599, 8.680995183118574253524087588320, 8.878467033920049179090364089141, 9.452820608061107156971105416785, 9.564040056767187668964397568384, 10.45778215425951597705850617163, 10.62504155870259967431267237679
