L(s) = 1 | + 4·13-s − 4·37-s − 4·61-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 4·13-s − 4·37-s − 4·61-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9008900305\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9008900305\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84516857620233318507649007436, −7.23657877521383497767263340020, −7.14117594131401513683028128746, −6.91575487484178863634416985091, −6.81527305640456196656582093577, −6.31594235116094141169306955317, −6.25480779584041687786407899601, −6.00712842047157907449913149768, −5.97765466754364041880612449728, −5.43722649342321044303429795478, −5.43626017265676189353926315350, −5.05874528787727002927089794351, −4.87568833691894492966308118606, −4.40429449059889297546425732176, −4.06236654040940025278172943283, −4.03831900037673161004832773549, −3.70082047242043809456201066502, −3.36019090097599852242217091277, −3.31422842149176062958162581066, −2.81946286797672009368096705447, −2.76129092971241327325370869272, −1.73880738035066004212755957453, −1.66274092463359388474131334946, −1.63729939982513489239532469311, −1.02178174975497537173849244892,
1.02178174975497537173849244892, 1.63729939982513489239532469311, 1.66274092463359388474131334946, 1.73880738035066004212755957453, 2.76129092971241327325370869272, 2.81946286797672009368096705447, 3.31422842149176062958162581066, 3.36019090097599852242217091277, 3.70082047242043809456201066502, 4.03831900037673161004832773549, 4.06236654040940025278172943283, 4.40429449059889297546425732176, 4.87568833691894492966308118606, 5.05874528787727002927089794351, 5.43626017265676189353926315350, 5.43722649342321044303429795478, 5.97765466754364041880612449728, 6.00712842047157907449913149768, 6.25480779584041687786407899601, 6.31594235116094141169306955317, 6.81527305640456196656582093577, 6.91575487484178863634416985091, 7.14117594131401513683028128746, 7.23657877521383497767263340020, 7.84516857620233318507649007436