| L(s) = 1 | + 2·5-s + 16·19-s − 25-s + 12·29-s + 16·31-s − 12·41-s − 2·49-s + 12·61-s − 32·71-s − 16·79-s − 20·89-s + 32·95-s + 28·101-s − 20·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 3.67·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 1.53·61-s − 3.79·71-s − 1.80·79-s − 2.11·89-s + 3.28·95-s + 2.78·101-s − 1.91·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.093211532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.093211532\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.926272091968759069212248150115, −8.486196106752036718806154990659, −8.365461418906253450800685106449, −7.83100285666323959724319695855, −7.36550396291301247274156863202, −7.14444368633372476856315203201, −6.67493755007643118678962109518, −6.20968661763707569030154193342, −6.00846662563900117861344616017, −5.30522118372809617491090806519, −5.28559974798868062265604710560, −4.74556533617142774785665008884, −4.38720648724227894710446407446, −3.75011043059342512679887158877, −3.02783462273235705876956197288, −2.92244067888758713257899134243, −2.62640613061990272688512021214, −1.51186729898671539299119058623, −1.35105777730234375396037589766, −0.71024289263720073803820861234,
0.71024289263720073803820861234, 1.35105777730234375396037589766, 1.51186729898671539299119058623, 2.62640613061990272688512021214, 2.92244067888758713257899134243, 3.02783462273235705876956197288, 3.75011043059342512679887158877, 4.38720648724227894710446407446, 4.74556533617142774785665008884, 5.28559974798868062265604710560, 5.30522118372809617491090806519, 6.00846662563900117861344616017, 6.20968661763707569030154193342, 6.67493755007643118678962109518, 7.14444368633372476856315203201, 7.36550396291301247274156863202, 7.83100285666323959724319695855, 8.365461418906253450800685106449, 8.486196106752036718806154990659, 8.926272091968759069212248150115
