L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 8-s + 3·9-s − 16-s + 3·17-s + 3·18-s − 2·19-s − 2·24-s − 2·25-s + 4·27-s + 3·34-s − 2·38-s − 2·41-s − 43-s − 2·48-s − 49-s − 2·50-s + 6·51-s + 4·54-s − 4·57-s + 2·59-s + 64-s − 3·67-s − 3·72-s + 2·73-s + ⋯ |
L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 8-s + 3·9-s − 16-s + 3·17-s + 3·18-s − 2·19-s − 2·24-s − 2·25-s + 4·27-s + 3·34-s − 2·38-s − 2·41-s − 43-s − 2·48-s − 49-s − 2·50-s + 6·51-s + 4·54-s − 4·57-s + 2·59-s + 64-s − 3·67-s − 3·72-s + 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.282510264\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.282510264\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.922741807300913451606231407558, −9.805546413331372149240328574686, −8.913984797699261443865254747969, −8.875759328095496419730436785667, −8.207638581912488640544356645592, −8.179365936341793739816386205435, −7.67843693285014608276004957411, −7.25719975699544822831324279846, −6.76580790788425478773297343984, −6.17120516767079900183398245249, −5.92556293128076132485729708758, −5.14911947558111937717449435008, −4.92090122364558420880363377253, −4.21864493318544513509819199347, −3.88253179760994641507756232081, −3.37475428530699469008270890243, −3.31315172622774955439834622362, −2.54743075305471823987166292303, −1.96363466395435767002066044616, −1.42634970793287883603469366590,
1.42634970793287883603469366590, 1.96363466395435767002066044616, 2.54743075305471823987166292303, 3.31315172622774955439834622362, 3.37475428530699469008270890243, 3.88253179760994641507756232081, 4.21864493318544513509819199347, 4.92090122364558420880363377253, 5.14911947558111937717449435008, 5.92556293128076132485729708758, 6.17120516767079900183398245249, 6.76580790788425478773297343984, 7.25719975699544822831324279846, 7.67843693285014608276004957411, 8.179365936341793739816386205435, 8.207638581912488640544356645592, 8.875759328095496419730436785667, 8.913984797699261443865254747969, 9.805546413331372149240328574686, 9.922741807300913451606231407558