| L(s) = 1 | + 18·3-s − 6·5-s + 98·7-s + 243·9-s + 90·11-s + 768·13-s − 108·15-s + 1.92e3·17-s − 2.24e3·19-s + 1.76e3·21-s − 6.35e3·23-s − 654·25-s + 2.91e3·27-s + 1.05e4·29-s + 3.31e3·31-s + 1.62e3·33-s − 588·35-s + 2.10e3·37-s + 1.38e4·39-s + 1.26e3·41-s + 5.76e3·43-s − 1.45e3·45-s − 1.56e4·47-s + 7.20e3·49-s + 3.46e4·51-s + 1.65e4·53-s − 540·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.107·5-s + 0.755·7-s + 9-s + 0.224·11-s + 1.26·13-s − 0.123·15-s + 1.61·17-s − 1.42·19-s + 0.872·21-s − 2.50·23-s − 0.209·25-s + 0.769·27-s + 2.33·29-s + 0.618·31-s + 0.258·33-s − 0.0811·35-s + 0.252·37-s + 1.45·39-s + 0.117·41-s + 0.475·43-s − 0.107·45-s − 1.03·47-s + 3/7·49-s + 1.86·51-s + 0.807·53-s − 0.0240·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.991669387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.991669387\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 138 p T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 90 T + 51246 T^{2} - 90 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 768 T + 689558 T^{2} - 768 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1926 T + 3761514 T^{2} - 1926 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2248 T + 3007830 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6354 T + 22960446 T^{2} + 6354 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10572 T + 60053694 T^{2} - 10572 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3312 T + 31130942 T^{2} - 3312 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2104 T + 14492118 T^{2} - 2104 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1266 T + 226048450 T^{2} - 1266 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5768 T - 18440058 T^{2} - 5768 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15612 T + 373470814 T^{2} + 15612 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16512 T + 599616358 T^{2} - 16512 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13140 T + 1459090998 T^{2} - 13140 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5796 T + 1309453982 T^{2} + 5796 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 56116 T + 2298831942 T^{2} - 56116 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11022 T - 822078402 T^{2} + 11022 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 85384 T + 5800143006 T^{2} + 85384 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19620 T + 6216467102 T^{2} - 19620 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 44424 T + 7801188630 T^{2} - 44424 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 211218 T + 21434789410 T^{2} - 211218 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 44864 T + 3170652414 T^{2} - 44864 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.73788599517428886758987050178, −10.44147114544742815034583253928, −9.852175836271121160601702220569, −9.838091991585157595872018825899, −8.714421872354062741163639907927, −8.579527667719131918440022482136, −8.283174240860898497130290318137, −7.77020517252972137557275260862, −7.44018738515136511795267339551, −6.47757871611835141016767793675, −6.20552494357699353835144655441, −5.72192341657863177969015414653, −4.62550956661090358858451257153, −4.50468082853022188551645698945, −3.57446630670126067340443351805, −3.51442178027130394397653100223, −2.42105519681349864822015371746, −2.03332029023314144094760677985, −1.25130220709984621670630565175, −0.68576481482159037703123855094,
0.68576481482159037703123855094, 1.25130220709984621670630565175, 2.03332029023314144094760677985, 2.42105519681349864822015371746, 3.51442178027130394397653100223, 3.57446630670126067340443351805, 4.50468082853022188551645698945, 4.62550956661090358858451257153, 5.72192341657863177969015414653, 6.20552494357699353835144655441, 6.47757871611835141016767793675, 7.44018738515136511795267339551, 7.77020517252972137557275260862, 8.283174240860898497130290318137, 8.579527667719131918440022482136, 8.714421872354062741163639907927, 9.838091991585157595872018825899, 9.852175836271121160601702220569, 10.44147114544742815034583253928, 10.73788599517428886758987050178
