| L(s) = 1 | + 4·5-s − 6·7-s − 9-s + 2·11-s − 8·13-s + 6·17-s − 6·19-s + 2·23-s + 11·25-s − 6·29-s − 24·35-s − 16·37-s − 12·43-s − 4·45-s − 10·47-s + 18·49-s + 8·55-s − 14·59-s + 18·61-s + 6·63-s − 32·65-s − 4·67-s + 16·71-s − 10·73-s − 12·77-s + 81-s + 24·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.26·7-s − 1/3·9-s + 0.603·11-s − 2.21·13-s + 1.45·17-s − 1.37·19-s + 0.417·23-s + 11/5·25-s − 1.11·29-s − 4.05·35-s − 2.63·37-s − 1.82·43-s − 0.596·45-s − 1.45·47-s + 18/7·49-s + 1.07·55-s − 1.82·59-s + 2.30·61-s + 0.755·63-s − 3.96·65-s − 0.488·67-s + 1.89·71-s − 1.17·73-s − 1.36·77-s + 1/9·81-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6242929082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6242929082\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.653976858942879259257762436987, −9.106309189021959827167912669434, −8.917837311603604779674577775645, −8.267205873821144939363906250142, −7.891379922770582023928293534658, −7.04856826438382263232025938415, −6.98010916421581563782250907528, −6.56191891001832729600629776126, −6.43181830637520079300615743134, −5.72088047262283426678519189245, −5.50326203304008999628408397645, −5.07952247830461095011450602086, −4.73433350620126477554795720053, −3.74902053988551720427260817106, −3.50844155819926030731595601368, −2.93736733401200392927571435665, −2.63394281614110867228576864515, −1.94714286290854385696877503072, −1.56146344371270900377637906142, −0.26541719914254131492948166175,
0.26541719914254131492948166175, 1.56146344371270900377637906142, 1.94714286290854385696877503072, 2.63394281614110867228576864515, 2.93736733401200392927571435665, 3.50844155819926030731595601368, 3.74902053988551720427260817106, 4.73433350620126477554795720053, 5.07952247830461095011450602086, 5.50326203304008999628408397645, 5.72088047262283426678519189245, 6.43181830637520079300615743134, 6.56191891001832729600629776126, 6.98010916421581563782250907528, 7.04856826438382263232025938415, 7.891379922770582023928293534658, 8.267205873821144939363906250142, 8.917837311603604779674577775645, 9.106309189021959827167912669434, 9.653976858942879259257762436987
