| L(s) = 1 | + 20·2-s + 144·4-s − 2.24e3·8-s + 3.13e3·9-s + 1.09e4·13-s − 8.16e4·16-s − 1.46e5·17-s + 6.27e4·18-s + 2.18e5·26-s − 2.56e5·29-s − 1.05e6·32-s − 2.92e6·34-s + 4.51e5·36-s + 6.94e6·37-s + 4.29e6·41-s + 9.57e6·49-s + 1.57e6·52-s − 1.64e6·53-s − 5.12e6·58-s − 2.94e7·61-s − 2.90e5·64-s − 2.10e7·68-s − 7.02e6·72-s + 1.14e7·73-s + 1.38e8·74-s − 3.31e7·81-s + 8.58e7·82-s + ⋯ |
| L(s) = 1 | + 5/4·2-s + 9/16·4-s − 0.546·8-s + 0.478·9-s + 0.383·13-s − 1.24·16-s − 1.75·17-s + 0.597·18-s + 0.478·26-s − 0.362·29-s − 1.01·32-s − 2.18·34-s + 0.269·36-s + 3.70·37-s + 1.51·41-s + 1.66·49-s + 0.215·52-s − 0.208·53-s − 0.453·58-s − 2.13·61-s − 0.0173·64-s − 0.984·68-s − 0.261·72-s + 0.403·73-s + 4.63·74-s − 0.771·81-s + 1.89·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.821758168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.821758168\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 - 5 p^{2} T + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 1046 p T^{2} + p^{16} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 195362 p^{2} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 87015362 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5470 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 73090 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 33587484482 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 100353384578 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 128222 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1701165473282 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3472030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2146882 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11766582970942 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10537750788862 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 824290 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 279781405698242 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14746078 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 579369070794818 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1290076545985922 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5725630 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1744457179595522 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1804713576833858 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 83324222 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 120619010 T + p^{8} T^{2} )^{2} \) |
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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
−13.06274448365495508289800426144, −12.16523050872682017951143637997, −11.54449094768855506138710161045, −10.91487486490400093165662042445, −10.88406627294641026297676485328, −9.597106663453854062282133004488, −9.457158896145852434135632415622, −8.769274987907796786415304895002, −8.079854756701911375413140979822, −7.35401758166011095200879046607, −6.74764006402317521654792801658, −6.02763873878261766002209968218, −5.79585884636010310450055239232, −4.76364072246339087921857912606, −4.14735060784961943484962439741, −4.09831564197085165340472669269, −2.75124450073670663198300714436, −2.52555440435739325971294378086, −1.39123682683298762205799344567, −0.45430277484204201231094552961,
0.45430277484204201231094552961, 1.39123682683298762205799344567, 2.52555440435739325971294378086, 2.75124450073670663198300714436, 4.09831564197085165340472669269, 4.14735060784961943484962439741, 4.76364072246339087921857912606, 5.79585884636010310450055239232, 6.02763873878261766002209968218, 6.74764006402317521654792801658, 7.35401758166011095200879046607, 8.079854756701911375413140979822, 8.769274987907796786415304895002, 9.457158896145852434135632415622, 9.597106663453854062282133004488, 10.88406627294641026297676485328, 10.91487486490400093165662042445, 11.54449094768855506138710161045, 12.16523050872682017951143637997, 13.06274448365495508289800426144
