| L(s) = 1 | + 16·2-s + 192·4-s + 2.04e3·8-s − 3.92e3·9-s − 1.08e4·11-s + 2.04e4·16-s − 6.27e4·18-s − 1.73e5·22-s − 3.31e4·23-s − 1.41e5·25-s − 2.07e5·29-s + 1.96e5·32-s − 7.53e5·36-s − 6.65e5·37-s − 1.30e6·43-s − 2.08e6·44-s − 5.30e5·46-s − 2.26e6·50-s + 1.11e6·53-s − 3.32e6·58-s + 1.83e6·64-s + 7.65e5·67-s − 3.77e6·71-s − 8.03e6·72-s − 1.06e7·74-s + 2.60e6·79-s + 1.06e7·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.79·9-s − 2.46·11-s + 5/4·16-s − 2.53·18-s − 3.48·22-s − 0.568·23-s − 1.81·25-s − 1.58·29-s + 1.06·32-s − 2.69·36-s − 2.16·37-s − 2.49·43-s − 3.69·44-s − 0.803·46-s − 2.56·50-s + 1.03·53-s − 2.23·58-s + 7/8·64-s + 0.311·67-s − 1.25·71-s − 2.53·72-s − 3.05·74-s + 0.594·79-s + 2.21·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 436 p^{2} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 141458 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 494 p T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 117368522 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 815304704 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1611436836 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 16580 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 103766 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 25781595850 T^{2} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 332798 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 221598586960 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 651334 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 494678271274 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 559046 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4671442429700 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2907900568158 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 382884 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 1886652 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2259485255472 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1301660 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 37649542741172 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 38536982158880 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15655166237184 T^{2} + p^{14} T^{4} \) |
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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
−12.08464134056486523977709925087, −12.01835585477243947674462968129, −11.12779205398376973062808663297, −11.07390455223881031314011555889, −10.12785379652450962478699252085, −9.943756576350319389840820375384, −8.592364029166140434624860525155, −8.392437496847322009558869223985, −7.57990510157511274344470844836, −7.23998677731400710555401505004, −6.02951021720362598339110115738, −5.80783966445415735000922907928, −5.21828982832556815502533774494, −4.82054485121698162356792941574, −3.45670566843168486575146783959, −3.37787803564035287723854883575, −2.27008540458185843506237061570, −2.03541501753607254571924680149, 0, 0,
2.03541501753607254571924680149, 2.27008540458185843506237061570, 3.37787803564035287723854883575, 3.45670566843168486575146783959, 4.82054485121698162356792941574, 5.21828982832556815502533774494, 5.80783966445415735000922907928, 6.02951021720362598339110115738, 7.23998677731400710555401505004, 7.57990510157511274344470844836, 8.392437496847322009558869223985, 8.592364029166140434624860525155, 9.943756576350319389840820375384, 10.12785379652450962478699252085, 11.07390455223881031314011555889, 11.12779205398376973062808663297, 12.01835585477243947674462968129, 12.08464134056486523977709925087
