L(s) = 1 | + 4-s + 2·19-s − 2·31-s − 2·49-s − 2·61-s − 64-s + 2·76-s + 2·79-s + 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯ |
L(s) = 1 | + 4-s + 2·19-s − 2·31-s − 2·49-s − 2·61-s − 64-s + 2·76-s + 2·79-s + 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.078107592\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078107592\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.90492307991078464450421584878, −10.69995353470028601555407630088, −10.04233191321693192879368578912, −9.585978252600445084794369590346, −9.293931448809770347876976610029, −8.902673805224168261443791993009, −8.187685716621138649985745615138, −7.77029512094824187863475239894, −7.34647399746674689781816656542, −7.10421018651269618562981977934, −6.54187402819094240718011622585, −5.96035021983221161950886139058, −5.70846878066531994409813910002, −4.82518356472865274244022063026, −4.81568163406284992066207298275, −3.58512094867504956576191343752, −3.44220510745738028070359012889, −2.73689939642412253442071093737, −1.99624806850556960155525512998, −1.38214971959858102173434535003,
1.38214971959858102173434535003, 1.99624806850556960155525512998, 2.73689939642412253442071093737, 3.44220510745738028070359012889, 3.58512094867504956576191343752, 4.81568163406284992066207298275, 4.82518356472865274244022063026, 5.70846878066531994409813910002, 5.96035021983221161950886139058, 6.54187402819094240718011622585, 7.10421018651269618562981977934, 7.34647399746674689781816656542, 7.77029512094824187863475239894, 8.187685716621138649985745615138, 8.902673805224168261443791993009, 9.293931448809770347876976610029, 9.585978252600445084794369590346, 10.04233191321693192879368578912, 10.69995353470028601555407630088, 10.90492307991078464450421584878