| L(s) = 1 | + 4·5-s − 6·11-s + 16·19-s + 11·25-s − 2·29-s − 4·31-s + 12·41-s − 49-s − 24·55-s − 20·59-s + 14·79-s + 16·89-s + 64·95-s + 24·101-s + 14·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.80·11-s + 3.67·19-s + 11/5·25-s − 0.371·29-s − 0.718·31-s + 1.87·41-s − 1/7·49-s − 3.23·55-s − 2.60·59-s + 1.57·79-s + 1.69·89-s + 6.56·95-s + 2.38·101-s + 1.34·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.268842460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268842460\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−9.855449712044065944039572953730, −9.435936388611970654287193969951, −9.185605115254482351151244510556, −9.035703162138487266246226417885, −8.132438033440835325101428994858, −7.72598046010930576964985762851, −7.44557662543581336673537201119, −7.26746247473276944102929865160, −6.38805764852827934882428425379, −6.06019852018519156107118067585, −5.62962321831618817269077185998, −5.30184409228837645111288328745, −5.00884605097615267223011452522, −4.61246984879818660047458870606, −3.48356144171216555923539033915, −3.21751479034159165091794271975, −2.65743497227812755452696957885, −2.20855245488941755405519631860, −1.47450250584823845651620228032, −0.803691589757017357547531986139,
0.803691589757017357547531986139, 1.47450250584823845651620228032, 2.20855245488941755405519631860, 2.65743497227812755452696957885, 3.21751479034159165091794271975, 3.48356144171216555923539033915, 4.61246984879818660047458870606, 5.00884605097615267223011452522, 5.30184409228837645111288328745, 5.62962321831618817269077185998, 6.06019852018519156107118067585, 6.38805764852827934882428425379, 7.26746247473276944102929865160, 7.44557662543581336673537201119, 7.72598046010930576964985762851, 8.132438033440835325101428994858, 9.035703162138487266246226417885, 9.185605115254482351151244510556, 9.435936388611970654287193969951, 9.855449712044065944039572953730
