L(s) = 1 | + 2-s + 2·4-s − 5-s − 5·7-s + 5·8-s − 10-s + 5·11-s − 10·13-s − 5·14-s + 5·16-s + 3·17-s − 19-s − 2·20-s + 5·22-s + 3·23-s + 5·25-s − 10·26-s − 10·28-s + 2·29-s + 10·32-s + 3·34-s + 5·35-s − 3·37-s − 38-s − 5·40-s + 10·41-s − 2·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 4-s − 0.447·5-s − 1.88·7-s + 1.76·8-s − 0.316·10-s + 1.50·11-s − 2.77·13-s − 1.33·14-s + 5/4·16-s + 0.727·17-s − 0.229·19-s − 0.447·20-s + 1.06·22-s + 0.625·23-s + 25-s − 1.96·26-s − 1.88·28-s + 0.371·29-s + 1.76·32-s + 0.514·34-s + 0.845·35-s − 0.493·37-s − 0.162·38-s − 0.790·40-s + 1.56·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.520953599\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.520953599\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97476396784659801462753665311, −10.56010096941720571124052028935, −10.01558233821006880516468028936, −9.715025815552146505131750347099, −9.459669099872577173280342956468, −8.914566785201781477898054519634, −8.112329697684099482089668531887, −7.65386135268362658060006331196, −7.10382399751362537956864056773, −6.91246022882598953942187620296, −6.63112735845138938000378784891, −6.07818742303336497276841721323, −5.17706966316837321509782203759, −5.03587339732697141324485714602, −4.11164406171929865931442361341, −4.00010692927557526776124361165, −3.02772887976224370827709106892, −2.79310863543652204176812187361, −2.00485062942749586867798591099, −0.816331244443204621885585140979,
0.816331244443204621885585140979, 2.00485062942749586867798591099, 2.79310863543652204176812187361, 3.02772887976224370827709106892, 4.00010692927557526776124361165, 4.11164406171929865931442361341, 5.03587339732697141324485714602, 5.17706966316837321509782203759, 6.07818742303336497276841721323, 6.63112735845138938000378784891, 6.91246022882598953942187620296, 7.10382399751362537956864056773, 7.65386135268362658060006331196, 8.112329697684099482089668531887, 8.914566785201781477898054519634, 9.459669099872577173280342956468, 9.715025815552146505131750347099, 10.01558233821006880516468028936, 10.56010096941720571124052028935, 10.97476396784659801462753665311