| L(s) = 1 | − 3-s − 11-s + 25-s + 27-s + 33-s − 3·41-s − 3·43-s + 49-s + 59-s + 3·67-s + 2·73-s − 75-s − 81-s + 2·83-s + 97-s − 2·107-s + 121-s + 3·123-s + 127-s + 3·129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3-s − 11-s + 25-s + 27-s + 33-s − 3·41-s − 3·43-s + 49-s + 59-s + 3·67-s + 2·73-s − 75-s − 81-s + 2·83-s + 97-s − 2·107-s + 121-s + 3·123-s + 127-s + 3·129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5542730866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5542730866\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−10.30401613582429353912948649270, −9.729923335212448247062228102154, −9.713698889122175401365338352956, −8.807684935307733772315429504973, −8.561260906443996324119220963297, −8.059808779249993604386062059198, −8.005446621875925014411523275044, −7.00871047801996196995493506003, −6.70989330109287694797106737161, −6.70528334974994246019662780858, −5.97533084442321874325234846217, −5.26936110194668502827172135590, −5.09349761546224666032753480224, −5.06196629728256267700134865641, −4.19236168335300638227054785059, −3.38229133065129211881688170383, −3.26355754649610131439298947658, −2.35311361942562411350292863515, −1.82502763960839879152377636775, −0.71063606592494478887874750811,
0.71063606592494478887874750811, 1.82502763960839879152377636775, 2.35311361942562411350292863515, 3.26355754649610131439298947658, 3.38229133065129211881688170383, 4.19236168335300638227054785059, 5.06196629728256267700134865641, 5.09349761546224666032753480224, 5.26936110194668502827172135590, 5.97533084442321874325234846217, 6.70528334974994246019662780858, 6.70989330109287694797106737161, 7.00871047801996196995493506003, 8.005446621875925014411523275044, 8.059808779249993604386062059198, 8.561260906443996324119220963297, 8.807684935307733772315429504973, 9.713698889122175401365338352956, 9.729923335212448247062228102154, 10.30401613582429353912948649270
