| L(s) = 1 | + 3·7-s − 6·11-s + 7·13-s − 6·19-s + 6·23-s − 2·25-s + 6·29-s + 12·41-s − 43-s − 49-s − 24·53-s − 6·59-s − 61-s − 15·67-s − 18·71-s − 18·77-s + 22·79-s − 12·89-s + 21·91-s − 9·97-s − 18·101-s + 2·103-s + 6·107-s + 6·113-s + 13·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.80·11-s + 1.94·13-s − 1.37·19-s + 1.25·23-s − 2/5·25-s + 1.11·29-s + 1.87·41-s − 0.152·43-s − 1/7·49-s − 3.29·53-s − 0.781·59-s − 0.128·61-s − 1.83·67-s − 2.13·71-s − 2.05·77-s + 2.47·79-s − 1.27·89-s + 2.20·91-s − 0.913·97-s − 1.79·101-s + 0.197·103-s + 0.580·107-s + 0.564·113-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213206038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213206038\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.575998133364500977744244022024, −8.761769351219909915723949317334, −8.692700063809163301382088294009, −8.185671717299670862931690995473, −7.85368742182022637492133138713, −7.77246511181238394803567399300, −7.12709110255618987692534609081, −6.48037458057486404128717187484, −6.32696931124475418608866006642, −5.68227205182762207979301500021, −5.57727699753901817016429502406, −4.69179727652537013997358929311, −4.65716645406775715395121822368, −4.28323666195234539858464983190, −3.48490803466939314477923148091, −2.91854700757476451901906785866, −2.72693299224450383401279500980, −1.71402981877806298870744535869, −1.55261980507020798552462442305, −0.55217117108562369530930829237,
0.55217117108562369530930829237, 1.55261980507020798552462442305, 1.71402981877806298870744535869, 2.72693299224450383401279500980, 2.91854700757476451901906785866, 3.48490803466939314477923148091, 4.28323666195234539858464983190, 4.65716645406775715395121822368, 4.69179727652537013997358929311, 5.57727699753901817016429502406, 5.68227205182762207979301500021, 6.32696931124475418608866006642, 6.48037458057486404128717187484, 7.12709110255618987692534609081, 7.77246511181238394803567399300, 7.85368742182022637492133138713, 8.185671717299670862931690995473, 8.692700063809163301382088294009, 8.761769351219909915723949317334, 9.575998133364500977744244022024
