L(s) = 1 | − 2·4-s + 3·16-s − 25-s + 2·37-s + 2·43-s − 4·64-s + 4·67-s + 4·79-s + 2·100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 2·4-s + 3·16-s − 25-s + 2·37-s + 2·43-s − 4·64-s + 4·67-s + 4·79-s + 2·100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9364221386\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9364221386\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.730356349045030904179477848138, −8.477871022431795773556003663855, −8.122830317066194019739345565571, −7.912239764178354488690315617026, −7.43366841098501318227454996021, −7.21374550682274048826263045610, −6.31611146052426385825735333878, −6.30882740193674643537076218907, −5.63008439754334195154618621950, −5.57306651088991432208630119969, −4.88573270404298101204509525849, −4.77721636568213698627598579167, −4.10985503883110089462179773535, −4.09363921636143933977865234583, −3.41014954310708957042753152505, −3.29365954359094964092295553037, −2.24873927156594603547048436836, −2.17288818483861411014490213707, −0.917085110099472635406967331121, −0.791363440723084137400474316082,
0.791363440723084137400474316082, 0.917085110099472635406967331121, 2.17288818483861411014490213707, 2.24873927156594603547048436836, 3.29365954359094964092295553037, 3.41014954310708957042753152505, 4.09363921636143933977865234583, 4.10985503883110089462179773535, 4.77721636568213698627598579167, 4.88573270404298101204509525849, 5.57306651088991432208630119969, 5.63008439754334195154618621950, 6.30882740193674643537076218907, 6.31611146052426385825735333878, 7.21374550682274048826263045610, 7.43366841098501318227454996021, 7.912239764178354488690315617026, 8.122830317066194019739345565571, 8.477871022431795773556003663855, 8.730356349045030904179477848138