| L(s) = 1 | + 4-s − 2·7-s − 8·11-s − 3·16-s − 10·17-s + 10·19-s + 2·23-s − 2·28-s − 4·31-s + 4·41-s − 2·43-s − 8·44-s − 8·47-s − 6·49-s − 6·53-s − 8·59-s − 7·64-s − 6·67-s − 10·68-s + 16·71-s + 4·73-s + 10·76-s + 16·77-s + 6·79-s + 8·83-s − 2·89-s + 2·92-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 2.41·11-s − 3/4·16-s − 2.42·17-s + 2.29·19-s + 0.417·23-s − 0.377·28-s − 0.718·31-s + 0.624·41-s − 0.304·43-s − 1.20·44-s − 1.16·47-s − 6/7·49-s − 0.824·53-s − 1.04·59-s − 7/8·64-s − 0.733·67-s − 1.21·68-s + 1.89·71-s + 0.468·73-s + 1.14·76-s + 1.82·77-s + 0.675·79-s + 0.878·83-s − 0.211·89-s + 0.208·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 174 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.82463167262979279694724553688, −7.68478877897372664592891740590, −7.43196996995358747366211413033, −6.86187893238974572079922039142, −6.66397888430848349963318502448, −6.34348417438734229098535940613, −5.93002667687888608312569214767, −5.33712232661972509256012598326, −5.16179081394574476551575418707, −4.77619728132496108747145644917, −4.53891000696561703427098557246, −3.82958618855256696754814448354, −3.26811208157143016797753365659, −3.06376280827926172525959248817, −2.65209461058203472766778692329, −2.16821535571126578535357402272, −1.90726276821360446601291293866, −1.00932586045321112289016426425, 0, 0,
1.00932586045321112289016426425, 1.90726276821360446601291293866, 2.16821535571126578535357402272, 2.65209461058203472766778692329, 3.06376280827926172525959248817, 3.26811208157143016797753365659, 3.82958618855256696754814448354, 4.53891000696561703427098557246, 4.77619728132496108747145644917, 5.16179081394574476551575418707, 5.33712232661972509256012598326, 5.93002667687888608312569214767, 6.34348417438734229098535940613, 6.66397888430848349963318502448, 6.86187893238974572079922039142, 7.43196996995358747366211413033, 7.68478877897372664592891740590, 7.82463167262979279694724553688
