| L(s) = 1 | − 3-s − 4-s − 2·5-s − 2·7-s + 12-s + 2·15-s + 16-s + 2·17-s + 8·19-s + 2·20-s + 2·21-s − 2·23-s − 25-s + 4·27-s + 2·28-s + 4·35-s − 8·37-s − 48-s − 2·49-s − 2·51-s − 8·57-s − 4·59-s − 2·60-s − 64-s − 2·68-s + 2·69-s − 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 1.83·19-s + 0.447·20-s + 0.436·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.377·28-s + 0.676·35-s − 1.31·37-s − 0.144·48-s − 2/7·49-s − 0.280·51-s − 1.05·57-s − 0.520·59-s − 0.258·60-s − 1/8·64-s − 0.242·68-s + 0.240·69-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.602450395332515902653177427899, −8.903226050669653133498406827043, −8.470972699616657219188598063610, −7.86248062458750721035094601342, −7.40099724198824325946742762396, −7.00468487896800689290681838843, −6.26051044940577238616952398244, −5.77860674375791231685655396763, −5.17227258685475493351488259492, −4.72567853542324858839362308362, −3.84279492885884900122908818342, −3.46561426917601934500160503796, −2.78287239495500909169658277300, −1.28286948279600013610537350332, 0,
1.28286948279600013610537350332, 2.78287239495500909169658277300, 3.46561426917601934500160503796, 3.84279492885884900122908818342, 4.72567853542324858839362308362, 5.17227258685475493351488259492, 5.77860674375791231685655396763, 6.26051044940577238616952398244, 7.00468487896800689290681838843, 7.40099724198824325946742762396, 7.86248062458750721035094601342, 8.470972699616657219188598063610, 8.903226050669653133498406827043, 9.602450395332515902653177427899
