| L(s) = 1 | + 2·5-s − 4·11-s − 4·13-s − 6·17-s − 4·19-s − 4·23-s − 25-s − 14·41-s + 16·43-s − 2·49-s − 14·53-s − 8·55-s + 20·59-s − 8·65-s + 8·67-s − 16·71-s + 20·73-s − 12·85-s − 14·89-s − 8·95-s + 24·97-s − 12·103-s − 24·107-s − 26·109-s + 2·113-s − 8·115-s + 8·121-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 2.18·41-s + 2.43·43-s − 2/7·49-s − 1.92·53-s − 1.07·55-s + 2.60·59-s − 0.992·65-s + 0.977·67-s − 1.89·71-s + 2.34·73-s − 1.30·85-s − 1.48·89-s − 0.820·95-s + 2.43·97-s − 1.18·103-s − 2.32·107-s − 2.49·109-s + 0.188·113-s − 0.746·115-s + 8/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4896193049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4896193049\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−9.403354953050472021151326550751, −8.714643589281107022118476768022, −8.454206172094154496293847256321, −7.987995535761772296699561901988, −7.80176788345108393989739195689, −7.10239989747201930335872525748, −6.94096243617436793190261024410, −6.36469580675078037295355549205, −6.19972812232711990769992120306, −5.51908528253197586950895527864, −5.31068554410166384508582435772, −4.88074179607357154506094850001, −4.43277887331843215419273386289, −3.98665374920527011235211934228, −3.48103743900229471873431534780, −2.54223295771904326946625707746, −2.49148487826042495589801341394, −2.10716667318588914734657913779, −1.42964158107550622580020415858, −0.22194259395141884986768355630,
0.22194259395141884986768355630, 1.42964158107550622580020415858, 2.10716667318588914734657913779, 2.49148487826042495589801341394, 2.54223295771904326946625707746, 3.48103743900229471873431534780, 3.98665374920527011235211934228, 4.43277887331843215419273386289, 4.88074179607357154506094850001, 5.31068554410166384508582435772, 5.51908528253197586950895527864, 6.19972812232711990769992120306, 6.36469580675078037295355549205, 6.94096243617436793190261024410, 7.10239989747201930335872525748, 7.80176788345108393989739195689, 7.987995535761772296699561901988, 8.454206172094154496293847256321, 8.714643589281107022118476768022, 9.403354953050472021151326550751
