| L(s) = 1 | + 4·5-s − 8·11-s + 14·19-s + 11·25-s − 16·29-s + 12·31-s − 2·41-s − 11·49-s − 32·55-s − 6·59-s − 20·61-s − 8·71-s − 8·79-s − 36·89-s + 56·95-s + 12·101-s + 24·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 48·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.41·11-s + 3.21·19-s + 11/5·25-s − 2.97·29-s + 2.15·31-s − 0.312·41-s − 1.57·49-s − 4.31·55-s − 0.781·59-s − 2.56·61-s − 0.949·71-s − 0.900·79-s − 3.81·89-s + 5.74·95-s + 1.19·101-s + 2.29·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 3.85·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658767722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658767722\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.947137787877985891771570913860, −8.357812896801854752480663228058, −8.161017441399455717768131971822, −7.46466925131181425043059962738, −7.45216427777512174578570722263, −7.24277567080354671463052156775, −6.44429046507735316524429438879, −6.06576980139572232381467260702, −5.68962111685535430054148541777, −5.40071207170233715028396805373, −5.30542858505136464101172176621, −4.65933381136888410786771622689, −4.51087437057785503057708849028, −3.41165444435760949160487150610, −3.00550711491933329347810071062, −2.96927021186807641477948940295, −2.37694314774657893475348858799, −1.57497393220846331868952268350, −1.53306319333047932739004716543, −0.47923739264543321685523704449,
0.47923739264543321685523704449, 1.53306319333047932739004716543, 1.57497393220846331868952268350, 2.37694314774657893475348858799, 2.96927021186807641477948940295, 3.00550711491933329347810071062, 3.41165444435760949160487150610, 4.51087437057785503057708849028, 4.65933381136888410786771622689, 5.30542858505136464101172176621, 5.40071207170233715028396805373, 5.68962111685535430054148541777, 6.06576980139572232381467260702, 6.44429046507735316524429438879, 7.24277567080354671463052156775, 7.45216427777512174578570722263, 7.46466925131181425043059962738, 8.161017441399455717768131971822, 8.357812896801854752480663228058, 8.947137787877985891771570913860
