| L(s) = 1 | + 3-s + 4-s − 3·7-s + 9-s + 12-s − 8·13-s + 16-s + 13·19-s − 3·21-s − 4·25-s + 27-s − 3·28-s + 4·31-s + 36-s − 8·37-s − 8·39-s − 20·43-s + 48-s + 6·49-s − 8·52-s + 13·57-s − 11·61-s − 3·63-s + 64-s − 20·67-s + 73-s − 4·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 1.13·7-s + 1/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 2.98·19-s − 0.654·21-s − 4/5·25-s + 0.192·27-s − 0.566·28-s + 0.718·31-s + 1/6·36-s − 1.31·37-s − 1.28·39-s − 3.04·43-s + 0.144·48-s + 6/7·49-s − 1.10·52-s + 1.72·57-s − 1.40·61-s − 0.377·63-s + 1/8·64-s − 2.44·67-s + 0.117·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 463428 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463428 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 613 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−8.119801576858130919911762591402, −7.80495298057094302800827995245, −7.44676852308875331668572355607, −6.93595029907129843297109750137, −6.77627637336013561672959026340, −6.05127081571010996105908569479, −5.43759723619659820297239545273, −5.03511946434807418365416097882, −4.62408814364291150905251025530, −3.61510966827781484667744540530, −3.20831452204152417896729991203, −2.93313735619318303974816130852, −2.19327979196672456056879163347, −1.39016632750272389638439630954, 0,
1.39016632750272389638439630954, 2.19327979196672456056879163347, 2.93313735619318303974816130852, 3.20831452204152417896729991203, 3.61510966827781484667744540530, 4.62408814364291150905251025530, 5.03511946434807418365416097882, 5.43759723619659820297239545273, 6.05127081571010996105908569479, 6.77627637336013561672959026340, 6.93595029907129843297109750137, 7.44676852308875331668572355607, 7.80495298057094302800827995245, 8.119801576858130919911762591402
