| L(s) = 1 | − 6·5-s + 2·7-s − 3·11-s + 4·13-s − 22·19-s − 48·23-s − 25-s − 78·29-s + 32·31-s − 12·35-s − 68·37-s + 21·41-s − 61·43-s − 84·47-s + 49·49-s + 18·55-s + 87·59-s − 56·61-s − 24·65-s − 31·67-s + 130·73-s − 6·77-s + 38·79-s − 84·83-s + 8·91-s + 132·95-s + 115·97-s + ⋯ |
| L(s) = 1 | − 6/5·5-s + 2/7·7-s − 0.272·11-s + 4/13·13-s − 1.15·19-s − 2.08·23-s − 0.0399·25-s − 2.68·29-s + 1.03·31-s − 0.342·35-s − 1.83·37-s + 0.512·41-s − 1.41·43-s − 1.78·47-s + 49-s + 0.327·55-s + 1.47·59-s − 0.918·61-s − 0.369·65-s − 0.462·67-s + 1.78·73-s − 0.0779·77-s + 0.481·79-s − 1.01·83-s + 8/91·91-s + 1.38·95-s + 1.18·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3965867242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3965867242\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 48 T + 1297 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 78 T + 2869 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 32 T + 63 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 21 T + 1828 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 83 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 84 T + 4561 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 87 T + 6004 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 31 T - 3528 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 38 T - 4797 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.26729777791522814177571438926, −10.62984242773453025152318447053, −10.50549230912654410464125436995, −9.681065709897717414121305381873, −9.556503464431691274171656623625, −8.639429730139509900997513276479, −8.387144418714661811456246725751, −7.954255050877193134614937063482, −7.70296077305515925207392737910, −6.92994335423164758375952847471, −6.67602924381158365905257451129, −5.74335066414490553893817043936, −5.65299863450669884256096866883, −4.69866682685112917830712835267, −4.27974455117982819334682753525, −3.63839910365923229668053678112, −3.43523304839160430459058477642, −2.15135789090770523946487800359, −1.78781212594882174615545493339, −0.25613617057784455493930726229,
0.25613617057784455493930726229, 1.78781212594882174615545493339, 2.15135789090770523946487800359, 3.43523304839160430459058477642, 3.63839910365923229668053678112, 4.27974455117982819334682753525, 4.69866682685112917830712835267, 5.65299863450669884256096866883, 5.74335066414490553893817043936, 6.67602924381158365905257451129, 6.92994335423164758375952847471, 7.70296077305515925207392737910, 7.954255050877193134614937063482, 8.387144418714661811456246725751, 8.639429730139509900997513276479, 9.556503464431691274171656623625, 9.681065709897717414121305381873, 10.50549230912654410464125436995, 10.62984242773453025152318447053, 11.26729777791522814177571438926
