| L(s) = 1 | − 6·5-s + 36·11-s − 124·13-s − 114·17-s − 76·19-s − 24·23-s + 125·25-s − 108·29-s − 112·31-s + 178·37-s + 756·41-s − 344·43-s + 192·47-s − 402·53-s − 216·55-s − 396·59-s + 254·61-s + 744·65-s + 1.01e3·67-s − 1.68e3·71-s + 890·73-s − 80·79-s − 216·83-s + 684·85-s + 1.63e3·89-s + 456·95-s − 2.02e3·97-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 0.986·11-s − 2.64·13-s − 1.62·17-s − 0.917·19-s − 0.217·23-s + 25-s − 0.691·29-s − 0.648·31-s + 0.790·37-s + 2.87·41-s − 1.21·43-s + 0.595·47-s − 1.04·53-s − 0.529·55-s − 0.873·59-s + 0.533·61-s + 1.41·65-s + 1.84·67-s − 2.80·71-s + 1.42·73-s − 0.113·79-s − 0.285·83-s + 0.872·85-s + 1.95·89-s + 0.492·95-s − 2.11·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.333235913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333235913\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 114 T + 8083 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 p T - 3 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 24 T - 11591 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 112 T - 17247 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 178 T - 18969 T^{2} - 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 192 T - 66959 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 402 T + 12727 T^{2} + 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 396 T - 48563 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 254 T - 162465 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1012 T + 723381 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 890 T + 403083 T^{2} - 890 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 80 T - 486639 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1638 T + 1978075 T^{2} - 1638 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1010 T + p^{3} T^{2} )^{2} \) |
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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
−9.029255558592590997349599013697, −8.923497652162725536121609070862, −8.377416197079429333854448799420, −7.82001890065565460814948374560, −7.52185868061613281143560104556, −7.23050638232465351042308915128, −6.69320365203738603565965598293, −6.57334375772520202614523992490, −5.91861075097303946200358644003, −5.52363707822920626694777538804, −4.74411503840821895583870477903, −4.67221805095848334298301244823, −4.23843192129614131371634627072, −3.88302496441562025885601956134, −3.06001763876551630952708410185, −2.70456784454764596783976509523, −2.01840154608359979620468136520, −1.92024308705085918366006901710, −0.73809328277203389995923891327, −0.33109818455107951230742245447,
0.33109818455107951230742245447, 0.73809328277203389995923891327, 1.92024308705085918366006901710, 2.01840154608359979620468136520, 2.70456784454764596783976509523, 3.06001763876551630952708410185, 3.88302496441562025885601956134, 4.23843192129614131371634627072, 4.67221805095848334298301244823, 4.74411503840821895583870477903, 5.52363707822920626694777538804, 5.91861075097303946200358644003, 6.57334375772520202614523992490, 6.69320365203738603565965598293, 7.23050638232465351042308915128, 7.52185868061613281143560104556, 7.82001890065565460814948374560, 8.377416197079429333854448799420, 8.923497652162725536121609070862, 9.029255558592590997349599013697
