| L(s) = 1 | + 4-s + 8·7-s − 12·19-s − 2·25-s + 8·28-s + 30·31-s + 10·37-s + 4·43-s + 34·49-s − 6·61-s − 64-s + 26·67-s − 24·73-s − 12·76-s + 14·79-s − 2·100-s + 18·103-s + 22·109-s − 22·121-s + 30·124-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 10·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.02·7-s − 2.75·19-s − 2/5·25-s + 1.51·28-s + 5.38·31-s + 1.64·37-s + 0.609·43-s + 34/7·49-s − 0.768·61-s − 1/8·64-s + 3.17·67-s − 2.80·73-s − 1.37·76-s + 1.57·79-s − 1/5·100-s + 1.77·103-s + 2.10·109-s − 2·121-s + 2.69·124-s + 0.0887·127-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 0.821·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.234080266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.234080266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 14 T^{2} - 93 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 191 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.202689111128458155127942555565, −8.062937273543248184158624971988, −7.69460532623000542606919127339, −7.69234716995406049631860252837, −7.53131984968944942787201571993, −6.83376050707930871656076506757, −6.68313317549482915516882156436, −6.52883763411019171637692746831, −6.14358191002135666671858986479, −6.05713420244159721423883155180, −5.84476978701006895763764233670, −5.10554786030703854449284809218, −5.00349061360399577560535568674, −4.96991472677130897186647312472, −4.35408400871195712251872878418, −4.30859774138697785308516809548, −4.27497606651134838330354698571, −3.87376712378304521646930159656, −3.05607118026959046866180109582, −2.84526195761167167651731042670, −2.26684014634673027421182729923, −2.26600465477775926159788286894, −1.92462144607653299217714762528, −1.06307045667549204768151036466, −1.05379719742946092856984198415,
1.05379719742946092856984198415, 1.06307045667549204768151036466, 1.92462144607653299217714762528, 2.26600465477775926159788286894, 2.26684014634673027421182729923, 2.84526195761167167651731042670, 3.05607118026959046866180109582, 3.87376712378304521646930159656, 4.27497606651134838330354698571, 4.30859774138697785308516809548, 4.35408400871195712251872878418, 4.96991472677130897186647312472, 5.00349061360399577560535568674, 5.10554786030703854449284809218, 5.84476978701006895763764233670, 6.05713420244159721423883155180, 6.14358191002135666671858986479, 6.52883763411019171637692746831, 6.68313317549482915516882156436, 6.83376050707930871656076506757, 7.53131984968944942787201571993, 7.69234716995406049631860252837, 7.69460532623000542606919127339, 8.062937273543248184158624971988, 8.202689111128458155127942555565
