| L(s) = 1 | − 2·3-s + 3·9-s + 4·19-s − 10·27-s + 40·43-s + 28·49-s − 8·57-s + 28·67-s − 4·73-s + 20·81-s − 40·97-s − 14·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s − 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 12·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.917·19-s − 1.92·27-s + 6.09·43-s + 4·49-s − 1.05·57-s + 3.42·67-s − 0.468·73-s + 20/9·81-s − 4.06·97-s − 1.27·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.61·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.415536525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415536525\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.48248092543818157325582404555, −5.96122972579934565921731206447, −5.82580812182862020977746479916, −5.62607484250304244258037464479, −5.60503327659787467048719508317, −5.52433909801697726038225359982, −5.25162710587691776601803982486, −5.01367039147918925453177322656, −4.67915416229772223847771036246, −4.33066385167820391298854058513, −4.21491379731949139334905176191, −4.01364874647016051935317989451, −3.92101143082692565849744152788, −3.82195191732713687406984003218, −3.40987808568640235787289625386, −2.89036653524585394325711291403, −2.79789330247872569365582623185, −2.54368752416887717798543260054, −2.30359772579287131400239658486, −2.08535902990997626546891535378, −1.73931510675138986527278114826, −1.06238918846506052376345789870, −1.00251626289012778177988572536, −0.882235799625464597329887579504, −0.32940969379534012830409059539,
0.32940969379534012830409059539, 0.882235799625464597329887579504, 1.00251626289012778177988572536, 1.06238918846506052376345789870, 1.73931510675138986527278114826, 2.08535902990997626546891535378, 2.30359772579287131400239658486, 2.54368752416887717798543260054, 2.79789330247872569365582623185, 2.89036653524585394325711291403, 3.40987808568640235787289625386, 3.82195191732713687406984003218, 3.92101143082692565849744152788, 4.01364874647016051935317989451, 4.21491379731949139334905176191, 4.33066385167820391298854058513, 4.67915416229772223847771036246, 5.01367039147918925453177322656, 5.25162710587691776601803982486, 5.52433909801697726038225359982, 5.60503327659787467048719508317, 5.62607484250304244258037464479, 5.82580812182862020977746479916, 5.96122972579934565921731206447, 6.48248092543818157325582404555
