| L(s) = 1 | − 4·4-s + 72·11-s + 16·16-s + 182·19-s − 552·29-s + 382·31-s + 120·41-s − 288·44-s + 565·49-s − 1.48e3·59-s + 334·61-s − 64·64-s + 1.17e3·71-s − 728·76-s − 328·79-s − 2.49e3·89-s + 888·101-s + 1.93e3·109-s + 2.20e3·116-s + 1.22e3·121-s − 1.52e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.97·11-s + 1/4·16-s + 2.19·19-s − 3.53·29-s + 2.21·31-s + 0.457·41-s − 0.986·44-s + 1.64·49-s − 3.28·59-s + 0.701·61-s − 1/8·64-s + 1.96·71-s − 1.09·76-s − 0.467·79-s − 2.97·89-s + 0.874·101-s + 1.69·109-s + 1.76·116-s + 0.921·121-s − 1.10·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.917707928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917707928\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 565 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4105 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9682 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20734 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 276 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 191 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156613 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 152354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 76790 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 744 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 167 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 392677 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 162866 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 659158 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1248 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 617545 T^{2} + p^{6} T^{4} \) |
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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
−11.03676595750601602421960513114, −10.41444289713960304349920412297, −9.684943289972472120030113041336, −9.558554573110323266582884314677, −9.213029109294961420954972781915, −8.853154039405335026580770775405, −8.138031921974847038396846961291, −7.63847244787795482520151454346, −7.30772988916306172683732525207, −6.72589469788591103074238166683, −6.21624627263818124653864531713, −5.47930591026541785530747225974, −5.45920473325252442351551552510, −4.28521682891484267071361517009, −4.25504169345876496702457517403, −3.42994989456467291157872212742, −3.05582752121864190127329327028, −1.92037612102120884323483971948, −1.27266691737280758281034634198, −0.62836190107926876370891687076,
0.62836190107926876370891687076, 1.27266691737280758281034634198, 1.92037612102120884323483971948, 3.05582752121864190127329327028, 3.42994989456467291157872212742, 4.25504169345876496702457517403, 4.28521682891484267071361517009, 5.45920473325252442351551552510, 5.47930591026541785530747225974, 6.21624627263818124653864531713, 6.72589469788591103074238166683, 7.30772988916306172683732525207, 7.63847244787795482520151454346, 8.138031921974847038396846961291, 8.853154039405335026580770775405, 9.213029109294961420954972781915, 9.558554573110323266582884314677, 9.684943289972472120030113041336, 10.41444289713960304349920412297, 11.03676595750601602421960513114
