| L(s) = 1 | + 8·7-s − 4·9-s − 8·17-s + 8·23-s − 2·25-s + 16·31-s + 4·41-s + 34·49-s − 32·63-s − 8·71-s − 8·73-s − 16·79-s + 7·81-s + 24·89-s − 8·97-s + 24·103-s + 28·113-s − 64·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 4/3·9-s − 1.94·17-s + 1.66·23-s − 2/5·25-s + 2.87·31-s + 0.624·41-s + 34/7·49-s − 4.03·63-s − 0.949·71-s − 0.936·73-s − 1.80·79-s + 7/9·81-s + 2.54·89-s − 0.812·97-s + 2.36·103-s + 2.63·113-s − 5.86·119-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.58·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.297820698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297820698\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−11.16864900599586194902434800352, −10.91209404595769029210519404696, −10.42637178033083658237172163282, −9.883314653812860460528287596748, −8.936781326194524102910309356880, −8.813439232030150242587399693514, −8.487612586958332116151505354964, −8.167058260342329260108230696044, −7.41511515077245583457173870416, −7.39445193201164554674056408186, −6.24521647187241645309584533271, −6.23489109353507760925346992833, −5.20880474696415961785815476276, −5.02210215319635069469174047645, −4.49960645185009216472236312771, −4.23899392513018733142803174210, −3.00436965059298781127362087828, −2.48954608600981803648358007327, −1.85171982909904310984154993292, −0.976243472162187210614835272268,
0.976243472162187210614835272268, 1.85171982909904310984154993292, 2.48954608600981803648358007327, 3.00436965059298781127362087828, 4.23899392513018733142803174210, 4.49960645185009216472236312771, 5.02210215319635069469174047645, 5.20880474696415961785815476276, 6.23489109353507760925346992833, 6.24521647187241645309584533271, 7.39445193201164554674056408186, 7.41511515077245583457173870416, 8.167058260342329260108230696044, 8.487612586958332116151505354964, 8.813439232030150242587399693514, 8.936781326194524102910309356880, 9.883314653812860460528287596748, 10.42637178033083658237172163282, 10.91209404595769029210519404696, 11.16864900599586194902434800352
