L(s) = 1 | + 2·3-s − 4·4-s + 3·5-s − 8·7-s + 3·9-s − 8·12-s + 6·15-s + 12·16-s − 17-s − 2·19-s − 12·20-s − 16·21-s + 18·23-s + 4·25-s + 4·27-s + 32·28-s − 24·35-s − 12·36-s − 8·37-s + 9·45-s + 24·48-s + 34·49-s − 2·51-s − 4·57-s + 12·59-s − 24·60-s − 24·63-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 2·4-s + 1.34·5-s − 3.02·7-s + 9-s − 2.30·12-s + 1.54·15-s + 3·16-s − 0.242·17-s − 0.458·19-s − 2.68·20-s − 3.49·21-s + 3.75·23-s + 4/5·25-s + 0.769·27-s + 6.04·28-s − 4.05·35-s − 2·36-s − 1.31·37-s + 1.34·45-s + 3.46·48-s + 34/7·49-s − 0.280·51-s − 0.529·57-s + 1.56·59-s − 3.09·60-s − 3.02·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1105425 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1105425 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444174087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444174087\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 17 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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
−8.569700646198540476002149945968, −7.63075451299586448757437920341, −7.08199582321199217163758748774, −6.66463575826783700250676263362, −6.51609740369614917462879752898, −5.78200824128863401364705700099, −5.20289650874661325264421481027, −5.14306131647545403565477997005, −4.22172406081991876402843560872, −3.81905951450552627127186117098, −3.25257599736447137754746076902, −3.00632648437536992753290178800, −2.56940330683685764435785210432, −1.37557520782302937993720081395, −0.55449800498610258560594445237,
0.55449800498610258560594445237, 1.37557520782302937993720081395, 2.56940330683685764435785210432, 3.00632648437536992753290178800, 3.25257599736447137754746076902, 3.81905951450552627127186117098, 4.22172406081991876402843560872, 5.14306131647545403565477997005, 5.20289650874661325264421481027, 5.78200824128863401364705700099, 6.51609740369614917462879752898, 6.66463575826783700250676263362, 7.08199582321199217163758748774, 7.63075451299586448757437920341, 8.569700646198540476002149945968