| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 9-s − 12·13-s + 5·16-s + 8·17-s − 2·18-s + 10·19-s − 24·26-s + 6·32-s + 16·34-s − 3·36-s + 20·38-s + 18·43-s − 14·47-s + 5·49-s − 36·52-s + 18·53-s + 7·64-s + 26·67-s + 24·68-s − 4·72-s + 30·76-s + 81-s − 12·83-s + 36·86-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1/3·9-s − 3.32·13-s + 5/4·16-s + 1.94·17-s − 0.471·18-s + 2.29·19-s − 4.70·26-s + 1.06·32-s + 2.74·34-s − 1/2·36-s + 3.24·38-s + 2.74·43-s − 2.04·47-s + 5/7·49-s − 4.99·52-s + 2.47·53-s + 7/8·64-s + 3.17·67-s + 2.91·68-s − 0.471·72-s + 3.44·76-s + 1/9·81-s − 1.31·83-s + 3.88·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.408054167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.408054167\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.472920182347050558552884558554, −8.645896187821273084535254444712, −7.969631931811865588266824721288, −7.83178345451926686537881847253, −7.33586029336778662182848311392, −7.19962048208332386724198551873, −6.99963361266531680512078253392, −6.25216350256709036178357979850, −5.68505001059782549990682148803, −5.44940470371143522397763335782, −5.21316461539423496443591414210, −4.92320036226580917283048308810, −4.42494578338344676080449938099, −3.76816583970952072355976908876, −3.52929594995818854568787566304, −2.80928748025778147989193326462, −2.62505262839079392557143865716, −2.28443905997124351215701378857, −1.32265765805017790193031875770, −0.68031127513285504461387104353,
0.68031127513285504461387104353, 1.32265765805017790193031875770, 2.28443905997124351215701378857, 2.62505262839079392557143865716, 2.80928748025778147989193326462, 3.52929594995818854568787566304, 3.76816583970952072355976908876, 4.42494578338344676080449938099, 4.92320036226580917283048308810, 5.21316461539423496443591414210, 5.44940470371143522397763335782, 5.68505001059782549990682148803, 6.25216350256709036178357979850, 6.99963361266531680512078253392, 7.19962048208332386724198551873, 7.33586029336778662182848311392, 7.83178345451926686537881847253, 7.969631931811865588266824721288, 8.645896187821273084535254444712, 9.472920182347050558552884558554
