| L(s) = 1 | − 15·4-s + 2·7-s − 10·13-s + 91·16-s − 42·19-s − 237·25-s − 30·28-s − 170·31-s − 288·37-s + 444·43-s − 1.26e3·49-s + 150·52-s − 1.05e3·61-s − 315·64-s − 790·67-s + 2.57e3·73-s + 630·76-s + 130·79-s − 20·91-s − 3.34e3·97-s + 3.55e3·100-s − 1.59e3·103-s + 1.71e3·109-s + 182·112-s + 2.55e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.87·4-s + 0.107·7-s − 0.213·13-s + 1.42·16-s − 0.507·19-s − 1.89·25-s − 0.202·28-s − 0.984·31-s − 1.27·37-s + 1.57·43-s − 3.69·49-s + 0.400·52-s − 2.20·61-s − 0.615·64-s − 1.44·67-s + 4.12·73-s + 0.950·76-s + 0.185·79-s − 0.0230·91-s − 3.49·97-s + 3.55·100-s − 1.52·103-s + 1.50·109-s + 0.153·112-s + 1.84·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 15 T^{2} + 67 p T^{4} + 15 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 237 T^{2} + 1472 p^{2} T^{4} + 237 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 636 T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 3144 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 12525 T^{2} + 85699064 T^{4} + 12525 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 21 T + 9758 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 2400 T^{2} - 202995106 T^{4} - 2400 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 90773 T^{2} + 3245998752 T^{4} + 90773 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 85 T + 46866 T^{2} + 85 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 144 T + 161 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 69603 T^{2} + 6607567592 T^{4} - 69603 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 222 T + 98774 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 123180 T^{2} + 4242436454 T^{4} + 123180 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 220805 T^{2} + 30443571504 T^{4} + 220805 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 303408 T^{2} + 40458951182 T^{4} + 303408 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 525 T + 454076 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 395 T + 578976 T^{2} + 395 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 346112 T^{2} + 264965405022 T^{4} + 346112 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1285 T + 1089084 T^{2} - 1285 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 65 T + 812214 T^{2} - 65 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 940748 T^{2} + 559932286614 T^{4} + 940748 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 1729493 T^{2} + 1736174150712 T^{4} + 1729493 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 835 T + p^{3} T^{2} )^{4} \) |
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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
−7.09563250631271784981379223052, −6.97918498967351266267986940560, −6.53497295814302583870548801487, −6.37140561445037966604542629544, −6.28681623737594517355371814437, −5.82164035764245080638883481781, −5.81780360099946531418955352465, −5.45129007588713451106638420577, −5.35581125275223033814100239403, −4.81937241994657514660668764859, −4.70596872151609167654741261228, −4.68653142342771184086027122179, −4.64906057700364780388889850587, −3.92588485957187384022219727553, −3.80215431549255673657814240932, −3.77619177206999412062606523496, −3.56000064439838314672243218183, −3.11878491577918737644228502153, −2.87498172998365056984112613672, −2.34671482132700046132034500675, −2.26204124196277821336886326489, −1.90739763328896539611176685055, −1.47415506250638776038473524036, −1.18239899652486859894939175402, −1.02901450623101708572377907410, 0, 0, 0, 0,
1.02901450623101708572377907410, 1.18239899652486859894939175402, 1.47415506250638776038473524036, 1.90739763328896539611176685055, 2.26204124196277821336886326489, 2.34671482132700046132034500675, 2.87498172998365056984112613672, 3.11878491577918737644228502153, 3.56000064439838314672243218183, 3.77619177206999412062606523496, 3.80215431549255673657814240932, 3.92588485957187384022219727553, 4.64906057700364780388889850587, 4.68653142342771184086027122179, 4.70596872151609167654741261228, 4.81937241994657514660668764859, 5.35581125275223033814100239403, 5.45129007588713451106638420577, 5.81780360099946531418955352465, 5.82164035764245080638883481781, 6.28681623737594517355371814437, 6.37140561445037966604542629544, 6.53497295814302583870548801487, 6.97918498967351266267986940560, 7.09563250631271784981379223052
