| L(s) = 1 | + 18·3-s + 36·5-s + 120·7-s + 243·9-s + 200·11-s + 284·13-s + 648·15-s + 2.67e3·17-s − 72·19-s + 2.16e3·21-s − 3.84e3·23-s + 2.65e3·25-s + 2.91e3·27-s + 1.02e4·29-s − 1.04e4·31-s + 3.60e3·33-s + 4.32e3·35-s + 1.31e4·37-s + 5.11e3·39-s + 4.16e3·41-s + 5.83e3·43-s + 8.74e3·45-s − 1.52e3·47-s − 1.48e4·49-s + 4.81e4·51-s + 9.01e3·53-s + 7.20e3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.643·5-s + 0.925·7-s + 9-s + 0.498·11-s + 0.466·13-s + 0.743·15-s + 2.24·17-s − 0.0457·19-s + 1.06·21-s − 1.51·23-s + 0.850·25-s + 0.769·27-s + 2.25·29-s − 1.96·31-s + 0.575·33-s + 0.596·35-s + 1.57·37-s + 0.538·39-s + 0.386·41-s + 0.481·43-s + 0.643·45-s − 0.100·47-s − 0.885·49-s + 2.59·51-s + 0.440·53-s + 0.320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.411855281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.411855281\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 36 T - 1362 T^{2} - 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 120 T + 29278 T^{2} - 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 200 T + 46406 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 284 T + 477054 T^{2} - 284 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2676 T + 4344262 T^{2} - 2676 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 4159894 T^{2} + 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3840 T + 9416686 T^{2} + 3840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10212 T + 64228638 T^{2} - 10212 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10488 T + 84559438 T^{2} + 10488 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13148 T + 125908974 T^{2} - 13148 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4164 T + 216207126 T^{2} - 4164 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5832 T + 168401542 T^{2} - 5832 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1520 T + 148144670 T^{2} + 1520 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9012 T + 816689646 T^{2} - 9012 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 55096 T + 1818478886 T^{2} - 55096 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 63444 T + 2677193342 T^{2} + 63444 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36792 T + 1148752246 T^{2} + 36792 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 37664 T + 2440628942 T^{2} + 37664 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37836 T + 2085902966 T^{2} + 37836 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 144888 T + 10711615534 T^{2} + 144888 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 109272 T + 10472055958 T^{2} - 109272 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 32556 T + 8836559958 T^{2} + 32556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 69092 T + 18339537030 T^{2} - 69092 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.31474362258981328664402008445, −12.91680540892338424814573793409, −12.10912481332933460893494155649, −11.94009229481645761521227229090, −11.04682137455119076810370795351, −10.42098276578784588577857983225, −9.913199136562547562677378010591, −9.546549543467672285070756495215, −8.626802183863672773537982939927, −8.505875212030996092364174976014, −7.54457031083119041491321289681, −7.50694669914831601736673824687, −6.24329802029815766213454453775, −5.84140299252974182624626090693, −4.90464331602273466484279221205, −4.17550519339418360688206196839, −3.37453894067809688386766629978, −2.61657655144675772129530448179, −1.60951289621691539682008742832, −1.09472059715368022048047907437,
1.09472059715368022048047907437, 1.60951289621691539682008742832, 2.61657655144675772129530448179, 3.37453894067809688386766629978, 4.17550519339418360688206196839, 4.90464331602273466484279221205, 5.84140299252974182624626090693, 6.24329802029815766213454453775, 7.50694669914831601736673824687, 7.54457031083119041491321289681, 8.505875212030996092364174976014, 8.626802183863672773537982939927, 9.546549543467672285070756495215, 9.913199136562547562677378010591, 10.42098276578784588577857983225, 11.04682137455119076810370795351, 11.94009229481645761521227229090, 12.10912481332933460893494155649, 12.91680540892338424814573793409, 13.31474362258981328664402008445
