| L(s) = 1 | + 2·3-s + 2·5-s − 9-s − 2·11-s + 4·15-s − 12·17-s + 4·19-s + 14·23-s + 25-s − 6·27-s + 16·29-s + 2·31-s − 4·33-s − 6·37-s + 4·43-s − 2·45-s + 8·47-s − 12·49-s − 24·51-s − 4·55-s + 8·57-s + 2·59-s + 6·67-s + 28·69-s + 2·71-s − 16·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 1/3·9-s − 0.603·11-s + 1.03·15-s − 2.91·17-s + 0.917·19-s + 2.91·23-s + 1/5·25-s − 1.15·27-s + 2.97·29-s + 0.359·31-s − 0.696·33-s − 0.986·37-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 1.71·49-s − 3.36·51-s − 0.539·55-s + 1.05·57-s + 0.260·59-s + 0.733·67-s + 3.37·69-s + 0.237·71-s − 1.87·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532405712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532405712\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 160 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 28 T + 354 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 171 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−9.567408487389932111274888326787, −9.250047570359225772207259155260, −8.858323423710327075881522831994, −8.723196679251823431227739172470, −8.265238748429276375857720329561, −8.006518168022008278395533337660, −7.10496106045072280103092405493, −7.05494180831234878844879154163, −6.45418485139458554603520270024, −6.32087699335111178126589373497, −5.44061169213042326395961858624, −5.18027318837337747713518095922, −4.64634246556385292204305254821, −4.42024099848878655319401885005, −3.46427121361801045713691588565, −2.99146649315298898635727298283, −2.57845887662117435563330480164, −2.46019208686065281319077729826, −1.58315353067553343859543527901, −0.70491255824819635924217735315,
0.70491255824819635924217735315, 1.58315353067553343859543527901, 2.46019208686065281319077729826, 2.57845887662117435563330480164, 2.99146649315298898635727298283, 3.46427121361801045713691588565, 4.42024099848878655319401885005, 4.64634246556385292204305254821, 5.18027318837337747713518095922, 5.44061169213042326395961858624, 6.32087699335111178126589373497, 6.45418485139458554603520270024, 7.05494180831234878844879154163, 7.10496106045072280103092405493, 8.006518168022008278395533337660, 8.265238748429276375857720329561, 8.723196679251823431227739172470, 8.858323423710327075881522831994, 9.250047570359225772207259155260, 9.567408487389932111274888326787
