L(s) = 1 | + 4-s − 6·9-s − 3·16-s + 10·25-s − 6·36-s − 28·49-s − 8·61-s − 7·64-s + 27·81-s + 10·100-s + 56·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 18·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 2·9-s − 3/4·16-s + 2·25-s − 36-s − 4·49-s − 1.02·61-s − 7/8·64-s + 3·81-s + 100-s + 5.36·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5814071280\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5814071280\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.26835583302356152021837947426, −11.21786645245511680209390523208, −10.66466370514893032831262775356, −10.50611649443584399182254907936, −10.23193054429604855466787590971, −9.652140086505478420732511583620, −9.384786331947788898180251109218, −9.031933621810800486731780558847, −8.710604288240186882615011007122, −8.680362817376058769151202987593, −7.997428208638764822405330422004, −7.84751197970826154652720172425, −7.62130075998172126003502019318, −6.75590916740167205077218524161, −6.64970921095770118350862487943, −6.55054750427678082702588075641, −5.91230804043502411888542382903, −5.61553303794249346366429241746, −5.11991821723020958559461200269, −4.71240468314306366469445297338, −4.44720620051296434479123125516, −3.34562197376954970251902214197, −3.21612165045091494677779403163, −2.71428527524336496712653239371, −1.91753482664858860475141963054,
1.91753482664858860475141963054, 2.71428527524336496712653239371, 3.21612165045091494677779403163, 3.34562197376954970251902214197, 4.44720620051296434479123125516, 4.71240468314306366469445297338, 5.11991821723020958559461200269, 5.61553303794249346366429241746, 5.91230804043502411888542382903, 6.55054750427678082702588075641, 6.64970921095770118350862487943, 6.75590916740167205077218524161, 7.62130075998172126003502019318, 7.84751197970826154652720172425, 7.997428208638764822405330422004, 8.680362817376058769151202987593, 8.710604288240186882615011007122, 9.031933621810800486731780558847, 9.384786331947788898180251109218, 9.652140086505478420732511583620, 10.23193054429604855466787590971, 10.50611649443584399182254907936, 10.66466370514893032831262775356, 11.21786645245511680209390523208, 11.26835583302356152021837947426