| L(s) = 1 | − 3-s + 2·4-s + 5-s + 5·7-s − 2·12-s − 2·13-s − 15-s − 6·17-s − 5·19-s + 2·20-s − 5·21-s − 6·23-s + 27-s + 10·28-s − 12·29-s − 5·31-s + 5·35-s + 7·37-s + 2·39-s + 24·41-s − 2·43-s − 6·47-s + 18·49-s + 6·51-s − 4·52-s + 5·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 0.447·5-s + 1.88·7-s − 0.577·12-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 1.14·19-s + 0.447·20-s − 1.09·21-s − 1.25·23-s + 0.192·27-s + 1.88·28-s − 2.22·29-s − 0.898·31-s + 0.845·35-s + 1.15·37-s + 0.320·39-s + 3.74·41-s − 0.304·43-s − 0.875·47-s + 18/7·49-s + 0.840·51-s − 0.554·52-s + 0.662·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192630213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192630213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−14.16852324351635364434656397988, −13.54708025636957339180612683556, −12.66404410653202593648386386208, −12.65215942826949931260768881590, −11.65439925390687154742767234976, −11.20947308788858526131724259271, −10.99301186095577513935888214513, −10.86881803606190678355921795499, −9.721217951859727755894324331464, −9.305993271327436186244625191105, −8.452643059318340487669814222981, −7.945762880910504350906558101369, −7.30805970911820392698975965539, −6.79790653633447899563705735539, −5.77032980731443694880549585483, −5.74119707570682436260227766143, −4.51482628306242117881742753993, −4.20104355949406265746614818739, −2.18183616664536748579387865347, −2.10177133252716354811089827771,
2.10177133252716354811089827771, 2.18183616664536748579387865347, 4.20104355949406265746614818739, 4.51482628306242117881742753993, 5.74119707570682436260227766143, 5.77032980731443694880549585483, 6.79790653633447899563705735539, 7.30805970911820392698975965539, 7.945762880910504350906558101369, 8.452643059318340487669814222981, 9.305993271327436186244625191105, 9.721217951859727755894324331464, 10.86881803606190678355921795499, 10.99301186095577513935888214513, 11.20947308788858526131724259271, 11.65439925390687154742767234976, 12.65215942826949931260768881590, 12.66404410653202593648386386208, 13.54708025636957339180612683556, 14.16852324351635364434656397988
