| L(s) = 1 | + 4·3-s − 4·4-s + 8·7-s + 4·9-s − 16·12-s − 40·13-s + 12·16-s + 32·19-s + 32·21-s − 10·25-s + 4·27-s − 32·28-s + 32·31-s − 16·36-s − 88·37-s − 160·39-s + 56·43-s + 48·48-s + 24·49-s + 160·52-s + 128·57-s − 64·61-s + 32·63-s − 32·64-s − 328·67-s + 200·73-s − 40·75-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 4-s + 8/7·7-s + 4/9·9-s − 4/3·12-s − 3.07·13-s + 3/4·16-s + 1.68·19-s + 1.52·21-s − 2/5·25-s + 4/27·27-s − 8/7·28-s + 1.03·31-s − 4/9·36-s − 2.37·37-s − 4.10·39-s + 1.30·43-s + 48-s + 0.489·49-s + 3.07·52-s + 2.24·57-s − 1.04·61-s + 0.507·63-s − 1/2·64-s − 4.89·67-s + 2.73·73-s − 0.533·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.083014864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083014864\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 4 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 220 T^{2} - 28218 T^{4} - 220 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1720 T^{2} + 1286322 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 44 T + 1782 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4060 T^{2} + 8942982 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 28 T + 3084 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 1064 T^{2} + 1942386 T^{4} + 1064 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10300 T^{2} + 42096102 T^{4} - 10300 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7300 T^{2} + 30092262 T^{4} - 7300 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 32 T + 6258 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 164 T + 14892 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 15124 T^{2} + 102823206 T^{4} - 15124 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 100 T + 11718 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 56 T + 12906 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26872 T^{2} + 275326098 T^{4} - 26872 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27940 T^{2} + 317327622 T^{4} - 27940 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 148 T + 22854 T^{2} - 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−12.55640238819559273788489887719, −12.38021451034777871471041853898, −11.91480053268635551782914030918, −11.77756303607672571941730686940, −11.62347478737633905942846947477, −10.67821091770215041182775861750, −10.52095346711521186251037321623, −10.14656069491292405038158354909, −9.735235018634811600124681359308, −9.504754017516068913340943387613, −9.047939057728980497652208854539, −8.831029818325252731278535938365, −8.477491156965453037419929654740, −7.943790537892119243599065353172, −7.53991075637317782693093146351, −7.42494431425121034409800706167, −7.17573344139437752004785015735, −6.18021632501698637963970249017, −5.60318173984488030937468040599, −4.97791719098989839396103759650, −4.79787416359626154393707868326, −4.44100400045966860534600906946, −3.37623278419340725389023178606, −2.94038497733243669636788563798, −2.05994636751125515591036685079,
2.05994636751125515591036685079, 2.94038497733243669636788563798, 3.37623278419340725389023178606, 4.44100400045966860534600906946, 4.79787416359626154393707868326, 4.97791719098989839396103759650, 5.60318173984488030937468040599, 6.18021632501698637963970249017, 7.17573344139437752004785015735, 7.42494431425121034409800706167, 7.53991075637317782693093146351, 7.943790537892119243599065353172, 8.477491156965453037419929654740, 8.831029818325252731278535938365, 9.047939057728980497652208854539, 9.504754017516068913340943387613, 9.735235018634811600124681359308, 10.14656069491292405038158354909, 10.52095346711521186251037321623, 10.67821091770215041182775861750, 11.62347478737633905942846947477, 11.77756303607672571941730686940, 11.91480053268635551782914030918, 12.38021451034777871471041853898, 12.55640238819559273788489887719
