| L(s) = 1 | + 16·2-s + 192·4-s + 250·5-s + 400·7-s + 2.04e3·8-s + 4.00e3·10-s + 3.12e3·11-s + 5.56e3·13-s + 6.40e3·14-s + 2.04e4·16-s + 2.11e4·17-s + 1.52e3·19-s + 4.80e4·20-s + 4.99e4·22-s + 4.62e4·23-s + 4.68e4·25-s + 8.89e4·26-s + 7.68e4·28-s + 1.66e5·29-s + 2.57e5·31-s + 1.96e5·32-s + 3.38e5·34-s + 1.00e5·35-s + 7.96e5·37-s + 2.44e4·38-s + 5.12e5·40-s − 5.85e5·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.440·7-s + 1.41·8-s + 1.26·10-s + 0.706·11-s + 0.701·13-s + 0.623·14-s + 5/4·16-s + 1.04·17-s + 0.0511·19-s + 1.34·20-s + 0.999·22-s + 0.791·23-s + 3/5·25-s + 0.992·26-s + 0.661·28-s + 1.26·29-s + 1.55·31-s + 1.06·32-s + 1.47·34-s + 0.394·35-s + 2.58·37-s + 0.0722·38-s + 1.26·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(13.53194349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.53194349\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 400 T + 10890 T^{2} - 400 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3120 T + 39731746 T^{2} - 3120 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5560 T + 25948890 T^{2} - 5560 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 21156 T + 262093030 T^{2} - 21156 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1528 T + 3096534 p^{2} T^{2} - 1528 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 46200 T - 870099506 T^{2} - 46200 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 166500 T + 36173764462 T^{2} - 166500 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 257872 T + 63436359918 T^{2} - 257872 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 796000 T + 336440515290 T^{2} - 796000 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 585840 T + 324453034162 T^{2} + 585840 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1544080 T + 1121553247878 T^{2} - 1544080 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1349400 T + 1464275840926 T^{2} + 1349400 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2630076 T + 4076065307518 T^{2} + 2630076 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 40560 T + 4721455727362 T^{2} + 40560 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 156724 T + 2212435689486 T^{2} - 156724 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1001600 T + 8509725179142 T^{2} + 1001600 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 851760 T + 9795115884958 T^{2} + 851760 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3988300 T + 21340535670294 T^{2} - 3988300 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2870912 T + 37678994136654 T^{2} + 2870912 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5108688 T + 34643927159590 T^{2} + 5108688 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5546520 T + 92136965639074 T^{2} + 5546520 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18054620 T + 238037349611910 T^{2} + 18054620 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.79647046583531362276718698016, −12.79153639817069560797336466722, −11.90198366603245626147321235641, −11.58036205335961705454783784857, −10.90420555236628732639048125885, −10.52284657024817751798948688243, −9.586837868294443224364952610967, −9.430545702682690441884879102394, −8.158077157689732582222540250286, −8.033143296589137234403955932285, −6.90750231659887069808909330115, −6.41197724130746397865194961254, −5.97178727370394865251829574628, −5.29562771327839153268364725023, −4.59409126807081470674029908011, −4.08522219961633942268594740401, −2.98756098093753851447242064459, −2.66905661903419266979423292546, −1.33641081939532362193456310478, −1.15631763372897469035747954187,
1.15631763372897469035747954187, 1.33641081939532362193456310478, 2.66905661903419266979423292546, 2.98756098093753851447242064459, 4.08522219961633942268594740401, 4.59409126807081470674029908011, 5.29562771327839153268364725023, 5.97178727370394865251829574628, 6.41197724130746397865194961254, 6.90750231659887069808909330115, 8.033143296589137234403955932285, 8.158077157689732582222540250286, 9.430545702682690441884879102394, 9.586837868294443224364952610967, 10.52284657024817751798948688243, 10.90420555236628732639048125885, 11.58036205335961705454783784857, 11.90198366603245626147321235641, 12.79153639817069560797336466722, 12.79647046583531362276718698016
