| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 5·16-s + 2·19-s + 6·20-s − 2·23-s + 3·25-s − 6·32-s − 4·38-s − 8·40-s + 4·46-s + 2·47-s + 49-s − 6·50-s + 2·53-s + 7·64-s + 6·76-s + 10·80-s − 6·92-s − 4·94-s + 4·95-s − 2·98-s + 9·100-s − 4·106-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 5·16-s + 2·19-s + 6·20-s − 2·23-s + 3·25-s − 6·32-s − 4·38-s − 8·40-s + 4·46-s + 2·47-s + 49-s − 6·50-s + 2·53-s + 7·64-s + 6·76-s + 10·80-s − 6·92-s − 4·94-s + 4·95-s − 2·98-s + 9·100-s − 4·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003784537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003784537\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.023133435735267180074635152507, −8.897890477540212442963696653360, −8.397375180912137666304748937216, −7.890029286948739667792957851102, −7.67849326775452767797140083068, −7.21853567200084208870066654996, −6.83242225115492755852226526122, −6.61138011113362149162249830833, −6.01154955992664242009573294743, −5.73557123231321272846006178503, −5.52496360970558548656805260817, −5.25536903506055819421618913054, −4.32749815004262421481440533971, −3.74303788260714063539041396806, −3.15290028292440029218077555315, −2.70217558363179904292694005913, −2.13504834424517437338992909118, −2.11926612017279930212291372171, −1.21051523781664732651206950632, −0.978875688099410759521335671866,
0.978875688099410759521335671866, 1.21051523781664732651206950632, 2.11926612017279930212291372171, 2.13504834424517437338992909118, 2.70217558363179904292694005913, 3.15290028292440029218077555315, 3.74303788260714063539041396806, 4.32749815004262421481440533971, 5.25536903506055819421618913054, 5.52496360970558548656805260817, 5.73557123231321272846006178503, 6.01154955992664242009573294743, 6.61138011113362149162249830833, 6.83242225115492755852226526122, 7.21853567200084208870066654996, 7.67849326775452767797140083068, 7.890029286948739667792957851102, 8.397375180912137666304748937216, 8.897890477540212442963696653360, 9.023133435735267180074635152507
