| L(s) = 1 | + 2·5-s + 4·7-s − 2·13-s − 4·17-s − 4·19-s + 4·23-s + 3·25-s + 4·29-s + 8·35-s + 4·37-s + 4·41-s + 8·47-s − 2·49-s + 8·59-s − 4·61-s − 4·65-s + 16·67-s − 16·73-s + 16·79-s + 16·83-s − 8·85-s + 4·89-s − 8·91-s − 8·95-s + 28·101-s + 16·103-s + 16·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.554·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 3/5·25-s + 0.742·29-s + 1.35·35-s + 0.657·37-s + 0.624·41-s + 1.16·47-s − 2/7·49-s + 1.04·59-s − 0.512·61-s − 0.496·65-s + 1.95·67-s − 1.87·73-s + 1.80·79-s + 1.75·83-s − 0.867·85-s + 0.423·89-s − 0.838·91-s − 0.820·95-s + 2.78·101-s + 1.57·103-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.849197709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.849197709\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−7.77178527368522457526115567791, −7.67716816961780512917323367726, −7.10968787679284758178749673787, −6.91007630187681357623891050701, −6.43717060689199147884568150547, −6.27381379429403619278212489631, −5.66499951296125852349587626701, −5.62788770923469182055400981431, −4.87545061940315412076964280988, −4.86702131929539501683163360965, −4.42629273822152145091973601414, −4.37721376004641010155950811242, −3.48311784552386331375692205206, −3.34067713114716632849164123389, −2.58848409745780847049653552218, −2.29287684528568941328194326822, −1.99159583294519917156577722064, −1.70846794109498457258813725282, −0.821435839594355957684711757980, −0.69218350628096647705185835349,
0.69218350628096647705185835349, 0.821435839594355957684711757980, 1.70846794109498457258813725282, 1.99159583294519917156577722064, 2.29287684528568941328194326822, 2.58848409745780847049653552218, 3.34067713114716632849164123389, 3.48311784552386331375692205206, 4.37721376004641010155950811242, 4.42629273822152145091973601414, 4.86702131929539501683163360965, 4.87545061940315412076964280988, 5.62788770923469182055400981431, 5.66499951296125852349587626701, 6.27381379429403619278212489631, 6.43717060689199147884568150547, 6.91007630187681357623891050701, 7.10968787679284758178749673787, 7.67716816961780512917323367726, 7.77178527368522457526115567791
