| L(s) = 1 | + 5·2-s − 18·3-s − 4-s + 58·5-s − 90·6-s + 286·7-s + 25·8-s + 243·9-s + 290·10-s + 18·12-s + 166·13-s + 1.43e3·14-s − 1.04e3·15-s − 73·16-s + 800·17-s + 1.21e3·18-s + 1.47e3·19-s − 58·20-s − 5.14e3·21-s − 3.37e3·23-s − 450·24-s + 698·25-s + 830·26-s − 2.91e3·27-s − 286·28-s − 6.60e3·29-s − 5.22e3·30-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 1.15·3-s − 0.0312·4-s + 1.03·5-s − 1.02·6-s + 2.20·7-s + 0.138·8-s + 9-s + 0.917·10-s + 0.0360·12-s + 0.272·13-s + 1.94·14-s − 1.19·15-s − 0.0712·16-s + 0.671·17-s + 0.883·18-s + 0.937·19-s − 0.0324·20-s − 2.54·21-s − 1.32·23-s − 0.159·24-s + 0.223·25-s + 0.240·26-s − 0.769·27-s − 0.0689·28-s − 1.45·29-s − 1.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.449431837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.449431837\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T + 13 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 58 T + 2666 T^{2} - 58 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 286 T + 49638 T^{2} - 286 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 166 T + 685578 T^{2} - 166 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 800 T + 2840414 T^{2} - 800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1476 T + 5490470 T^{2} - 1476 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3370 T + 9784358 T^{2} + 3370 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6600 T + 44401126 T^{2} + 6600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7528 T + 66189630 T^{2} + 7528 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 29916 T + 361611230 T^{2} + 29916 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5780 T + 217632230 T^{2} - 5780 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16656 T + 264340262 T^{2} - 16656 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7850 T + 191750726 T^{2} - 7850 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14178 T + 671305114 T^{2} - 14178 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17300 T + 1408269110 T^{2} - 17300 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2946 T + 1451133506 T^{2} - 2946 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 31336 T + 2492616438 T^{2} - 31336 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 33810 T + 3469543750 T^{2} + 33810 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60644 T + 2552552022 T^{2} + 60644 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1870 T + 2176543686 T^{2} + 1870 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 58296 T + 101571026 p T^{2} - 58296 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 92388 T + 3271275766 T^{2} - 92388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7120 T - 2888428386 T^{2} - 7120 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.86752830821271002037401979463, −10.62747249996840682021436115910, −10.06338697120673918867709152928, −9.637403593532933429531782819135, −8.784367233335181981984995726319, −8.757435635734643461696033989687, −7.76645257172123051135918296421, −7.36698870037065148560342998601, −7.21177792217797386139974358204, −6.17349193589250423788786064266, −5.61746914027455304337579506837, −5.51332440625329750973043016669, −5.03267020794150003241371454619, −4.70672983249106871783321159994, −3.80349605868615583409602666246, −3.70420307804801337837294864871, −2.18014954881080629809643886662, −1.72965327582353802102100806118, −1.48076218823276650986331412550, −0.48968916075945412115532132588,
0.48968916075945412115532132588, 1.48076218823276650986331412550, 1.72965327582353802102100806118, 2.18014954881080629809643886662, 3.70420307804801337837294864871, 3.80349605868615583409602666246, 4.70672983249106871783321159994, 5.03267020794150003241371454619, 5.51332440625329750973043016669, 5.61746914027455304337579506837, 6.17349193589250423788786064266, 7.21177792217797386139974358204, 7.36698870037065148560342998601, 7.76645257172123051135918296421, 8.757435635734643461696033989687, 8.784367233335181981984995726319, 9.637403593532933429531782819135, 10.06338697120673918867709152928, 10.62747249996840682021436115910, 10.86752830821271002037401979463
