| L(s) = 1 | + 4·13-s − 8·19-s + 5·25-s + 4·31-s + 10·37-s + 16·43-s − 14·61-s + 16·67-s + 10·73-s + 4·79-s + 28·97-s − 20·103-s − 2·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 1.83·19-s + 25-s + 0.718·31-s + 1.64·37-s + 2.43·43-s − 1.79·61-s + 1.95·67-s + 1.17·73-s + 0.450·79-s + 2.84·97-s − 1.97·103-s − 0.191·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527861701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527861701\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.327195826100525849810352730141, −9.142752585466131737628840223078, −8.638371561218495891672532847072, −8.426007348907668438143942109161, −7.88020534597555466468201777183, −7.69111512707236901276057619146, −7.06689096987763149485541235122, −6.60406626101844044642049746966, −6.19809804792488556972415853742, −6.10664854085932385858009365055, −5.51712518259139024214319348817, −4.93121025844297914527486626375, −4.32272852747450878338304975129, −4.31605028866775071339268285402, −3.61549607490624758598875153695, −3.13166595465221250917309521245, −2.41205460827750416779477760803, −2.18284295583693671543248892307, −1.20766030785818521395291901830, −0.67158045138158949210788812271,
0.67158045138158949210788812271, 1.20766030785818521395291901830, 2.18284295583693671543248892307, 2.41205460827750416779477760803, 3.13166595465221250917309521245, 3.61549607490624758598875153695, 4.31605028866775071339268285402, 4.32272852747450878338304975129, 4.93121025844297914527486626375, 5.51712518259139024214319348817, 6.10664854085932385858009365055, 6.19809804792488556972415853742, 6.60406626101844044642049746966, 7.06689096987763149485541235122, 7.69111512707236901276057619146, 7.88020534597555466468201777183, 8.426007348907668438143942109161, 8.638371561218495891672532847072, 9.142752585466131737628840223078, 9.327195826100525849810352730141
