| L(s) = 1 | + 2·4-s − 10·7-s + 2·13-s + 116·19-s + 22·25-s − 20·28-s + 20·31-s − 100·37-s − 28·43-s + 123·49-s + 4·52-s − 46·61-s − 8·64-s + 38·67-s − 388·73-s + 232·76-s − 154·79-s − 20·91-s + 98·97-s + 44·100-s + 326·103-s + 8·109-s − 170·121-s + 40·124-s + 127-s + 131-s − 1.16e3·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.42·7-s + 2/13·13-s + 6.10·19-s + 0.879·25-s − 5/7·28-s + 0.645·31-s − 2.70·37-s − 0.651·43-s + 2.51·49-s + 1/13·52-s − 0.754·61-s − 1/8·64-s + 0.567·67-s − 5.31·73-s + 3.05·76-s − 1.94·79-s − 0.219·91-s + 1.01·97-s + 0.439·100-s + 3.16·103-s + 0.0733·109-s − 1.40·121-s + 0.322·124-s + 0.00787·127-s + 0.00763·131-s − 8.72·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.785227041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785227041\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 22 T^{2} - 141 T^{4} - 22 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 170 T^{2} + 14259 T^{4} + 170 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 986 T^{2} + 692355 T^{4} + 986 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1394 T^{2} + 1235955 T^{4} + 1394 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T - 861 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 3074 T^{2} + 6623715 T^{4} + 3074 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T - 1653 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 4346 T^{2} + 14008035 T^{4} + 4346 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 1750 T^{2} - 9054861 T^{4} - 1750 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 23 T - 3192 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 19 T - 4128 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 77 T - 312 T^{2} + 77 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 334 T^{2} - 47346765 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 10010 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 49 T - 7008 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.269490158799491216141707582417, −9.014271242020913458951775813444, −8.813096840195227283944095371187, −8.524414648127735386752547788684, −7.85182493129889192832604739590, −7.71953122955539998898266217951, −7.32069222616890723249216268905, −7.31402114981213964124098180618, −7.12796704966421837072672758211, −6.68508586845698356336206502199, −6.42787180637193247668204617487, −6.01945073639271631169739804103, −5.56861421852791701842672904333, −5.45941698003918718007645857599, −5.24551323020402501043498642311, −4.92877923061377971168191209855, −4.34550843520151380277236248954, −3.88292612436839142850836556864, −3.37717430426346081193977805578, −3.14135425398315323102892716856, −2.96395057147542043121437165680, −2.79217742867388587121974580904, −1.59505310387997266944460933672, −1.33133852403946433233456482866, −0.61990618642850739388196062149,
0.61990618642850739388196062149, 1.33133852403946433233456482866, 1.59505310387997266944460933672, 2.79217742867388587121974580904, 2.96395057147542043121437165680, 3.14135425398315323102892716856, 3.37717430426346081193977805578, 3.88292612436839142850836556864, 4.34550843520151380277236248954, 4.92877923061377971168191209855, 5.24551323020402501043498642311, 5.45941698003918718007645857599, 5.56861421852791701842672904333, 6.01945073639271631169739804103, 6.42787180637193247668204617487, 6.68508586845698356336206502199, 7.12796704966421837072672758211, 7.31402114981213964124098180618, 7.32069222616890723249216268905, 7.71953122955539998898266217951, 7.85182493129889192832604739590, 8.524414648127735386752547788684, 8.813096840195227283944095371187, 9.014271242020913458951775813444, 9.269490158799491216141707582417
