| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 2·9-s + 2·11-s − 4·16-s + 4·18-s − 4·22-s − 12·23-s + 10·25-s − 16·29-s + 2·32-s − 2·36-s − 4·37-s + 40·43-s + 2·44-s + 24·46-s − 20·50-s − 4·53-s + 32·58-s + 3·64-s − 16·67-s + 64·71-s − 4·72-s + 8·74-s + 16·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s − 2/3·9-s + 0.603·11-s − 16-s + 0.942·18-s − 0.852·22-s − 2.50·23-s + 2·25-s − 2.97·29-s + 0.353·32-s − 1/3·36-s − 0.657·37-s + 6.09·43-s + 0.301·44-s + 3.53·46-s − 2.82·50-s − 0.549·53-s + 4.20·58-s + 3/8·64-s − 1.95·67-s + 7.59·71-s − 0.471·72-s + 0.929·74-s + 1.80·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.234735308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234735308\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 20 T^{2} + 39 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 12 T^{2} - 817 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 68 T^{2} + 2415 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^3$ | \( 1 - 24 T^{2} - 3145 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 74 T^{2} + 147 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23578761412582433242552160222, −7.00373500656771295472481831342, −6.58264945385010845091290977510, −6.45319301036509355692565872599, −6.16565291277193775863964397411, −6.00618605548551538378192941071, −5.81593142170049109768435747105, −5.65331695370207060271284626456, −5.13649040754469950502269705972, −5.06540838151111309884663631527, −4.96062900121633490672830171472, −4.36382061921744176456898733077, −4.32104686664175196371116493131, −3.93916196950605438771150927825, −3.70046234650284048089165966192, −3.60119681789291084263510779006, −3.38027428860066129907985008354, −2.74174831006288833916842200838, −2.52574081106356780936833481250, −2.11040485851714521394313151517, −1.97992678074676154169877537150, −1.85187184812690158034404391217, −0.905456057023445651646997909672, −0.835246760862546430766975005413, −0.47469420424988958869217618089,
0.47469420424988958869217618089, 0.835246760862546430766975005413, 0.905456057023445651646997909672, 1.85187184812690158034404391217, 1.97992678074676154169877537150, 2.11040485851714521394313151517, 2.52574081106356780936833481250, 2.74174831006288833916842200838, 3.38027428860066129907985008354, 3.60119681789291084263510779006, 3.70046234650284048089165966192, 3.93916196950605438771150927825, 4.32104686664175196371116493131, 4.36382061921744176456898733077, 4.96062900121633490672830171472, 5.06540838151111309884663631527, 5.13649040754469950502269705972, 5.65331695370207060271284626456, 5.81593142170049109768435747105, 6.00618605548551538378192941071, 6.16565291277193775863964397411, 6.45319301036509355692565872599, 6.58264945385010845091290977510, 7.00373500656771295472481831342, 7.23578761412582433242552160222
