| L(s) = 1 | + 3·4-s + 4·5-s − 4·7-s + 2·11-s − 4·13-s + 5·16-s + 2·17-s + 10·19-s + 12·20-s + 6·23-s + 11·25-s − 12·28-s + 10·31-s − 16·35-s + 14·41-s + 2·43-s + 6·44-s − 12·47-s − 2·49-s − 12·52-s − 10·53-s + 8·55-s − 14·59-s − 28·61-s + 3·64-s − 16·65-s + 6·68-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.78·5-s − 1.51·7-s + 0.603·11-s − 1.10·13-s + 5/4·16-s + 0.485·17-s + 2.29·19-s + 2.68·20-s + 1.25·23-s + 11/5·25-s − 2.26·28-s + 1.79·31-s − 2.70·35-s + 2.18·41-s + 0.304·43-s + 0.904·44-s − 1.75·47-s − 2/7·49-s − 1.66·52-s − 1.37·53-s + 1.07·55-s − 1.82·59-s − 3.58·61-s + 3/8·64-s − 1.98·65-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.605767968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.605767968\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.72440959888416029222835824497, −10.61288881307662799790860200276, −9.783042373734985910884675741064, −9.616126866341154029361737278570, −9.409181422224645400822259345420, −9.224970590422395118102282445120, −8.079399590540552084644177390121, −7.70169784541405369869938290406, −7.13348820787158648782267327938, −6.82581635358576888060922078876, −6.31056856910252442765294430321, −6.10826595620333019526598562043, −5.66923656596414643416684422466, −4.98326572817606007715464599997, −4.54027303025437624322113985333, −3.10210627727625880489298497705, −2.92509799788045075847693990026, −2.89748389829881884831127720787, −1.68767686535349208508781033229, −1.21101781460669823263287090664,
1.21101781460669823263287090664, 1.68767686535349208508781033229, 2.89748389829881884831127720787, 2.92509799788045075847693990026, 3.10210627727625880489298497705, 4.54027303025437624322113985333, 4.98326572817606007715464599997, 5.66923656596414643416684422466, 6.10826595620333019526598562043, 6.31056856910252442765294430321, 6.82581635358576888060922078876, 7.13348820787158648782267327938, 7.70169784541405369869938290406, 8.079399590540552084644177390121, 9.224970590422395118102282445120, 9.409181422224645400822259345420, 9.616126866341154029361737278570, 9.783042373734985910884675741064, 10.61288881307662799790860200276, 10.72440959888416029222835824497
