| L(s) = 1 | + 3-s − 5-s − 7-s − 4·11-s − 10·13-s − 15-s − 6·17-s + 5·19-s − 21-s − 2·23-s − 27-s − 4·29-s + 9·31-s − 4·33-s + 35-s + 11·37-s − 10·39-s − 16·41-s + 2·43-s − 6·47-s − 6·49-s − 6·51-s + 4·53-s + 4·55-s + 5·57-s − 6·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 1.20·11-s − 2.77·13-s − 0.258·15-s − 1.45·17-s + 1.14·19-s − 0.218·21-s − 0.417·23-s − 0.192·27-s − 0.742·29-s + 1.61·31-s − 0.696·33-s + 0.169·35-s + 1.80·37-s − 1.60·39-s − 2.49·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s − 0.840·51-s + 0.549·53-s + 0.539·55-s + 0.662·57-s − 0.781·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6354553690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6354553690\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−10.05560209275153186532644376495, −9.853597649926108105559691902079, −9.776934062001263769006570351772, −9.344092117923501891631402364703, −8.547839257550318905317780035063, −8.188853103986099453921521824006, −8.026270976654261497926891032613, −7.36057067234544849202461875420, −7.11928899337221849053216229209, −6.73945070528450399461826712417, −6.11371779384465494944115281814, −5.28220107255566818068513460552, −5.16074030653803085115883402911, −4.57711599249937384901525294009, −4.18068974634629887221300775252, −3.35244787926522943503636513056, −2.69809606306714530580035654118, −2.60447338402062283690164967308, −1.87152942536204832214286159543, −0.35059501255559014987693374454,
0.35059501255559014987693374454, 1.87152942536204832214286159543, 2.60447338402062283690164967308, 2.69809606306714530580035654118, 3.35244787926522943503636513056, 4.18068974634629887221300775252, 4.57711599249937384901525294009, 5.16074030653803085115883402911, 5.28220107255566818068513460552, 6.11371779384465494944115281814, 6.73945070528450399461826712417, 7.11928899337221849053216229209, 7.36057067234544849202461875420, 8.026270976654261497926891032613, 8.188853103986099453921521824006, 8.547839257550318905317780035063, 9.344092117923501891631402364703, 9.776934062001263769006570351772, 9.853597649926108105559691902079, 10.05560209275153186532644376495
