| L(s) = 1 | + 14·5-s + 24·7-s + 17·9-s − 32·11-s − 56·13-s + 154·17-s + 224·19-s − 68·23-s + 151·25-s − 472·29-s + 196·31-s + 336·35-s − 346·37-s − 840·41-s − 688·43-s + 238·45-s − 84·47-s + 338·49-s + 438·53-s − 448·55-s + 56·59-s − 98·61-s + 408·63-s − 784·65-s − 336·67-s + 1.79e3·71-s − 966·73-s + ⋯ |
| L(s) = 1 | + 1.25·5-s + 1.29·7-s + 0.629·9-s − 0.877·11-s − 1.19·13-s + 2.19·17-s + 2.70·19-s − 0.616·23-s + 1.20·25-s − 3.02·29-s + 1.13·31-s + 1.62·35-s − 1.53·37-s − 3.19·41-s − 2.43·43-s + 0.788·45-s − 0.260·47-s + 0.985·49-s + 1.13·53-s − 1.09·55-s + 0.123·59-s − 0.205·61-s + 0.815·63-s − 1.49·65-s − 0.612·67-s + 2.99·71-s − 1.54·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.285159441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285159441\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 34 p T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 17 T^{2} - 440 T^{4} - 17 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 14 T + 9 p T^{2} + 1386 T^{3} - 17324 T^{4} + 1386 p^{3} T^{5} + 9 p^{7} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 32 T - 81 T^{2} - 49824 T^{3} - 1913480 T^{4} - 49824 p^{3} T^{5} - 81 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 28 T + 3998 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 154 T + 8553 T^{2} - 821898 T^{3} + 89262292 T^{4} - 821898 p^{3} T^{5} + 8553 p^{6} T^{6} - 154 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 224 T + 24247 T^{2} - 2735264 T^{3} + 281109976 T^{4} - 2735264 p^{3} T^{5} + 24247 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 68 T - 19053 T^{2} - 44676 T^{3} + 356304232 T^{4} - 44676 p^{3} T^{5} - 19053 p^{6} T^{6} + 68 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 236 T + 33694 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 196 T - 28957 T^{2} - 1527036 T^{3} + 2507166392 T^{4} - 1527036 p^{3} T^{5} - 28957 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 346 T + 53749 T^{2} - 12227294 T^{3} - 4278055868 T^{4} - 12227294 p^{3} T^{5} + 53749 p^{6} T^{6} + 346 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 420 T + 176614 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 84 T - 194029 T^{2} - 551124 T^{3} + 28923386808 T^{4} - 551124 p^{3} T^{5} - 194029 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 438 T - 88603 T^{2} + 7580466 T^{3} + 27924999492 T^{4} + 7580466 p^{3} T^{5} - 88603 p^{6} T^{6} - 438 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 56 T - 357753 T^{2} + 2792664 T^{3} + 87416268136 T^{4} + 2792664 p^{3} T^{5} - 357753 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 98 T - 244147 T^{2} - 19620678 T^{3} + 10689270116 T^{4} - 19620678 p^{3} T^{5} - 244147 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 336 T - 515041 T^{2} + 8874096 T^{3} + 269891554152 T^{4} + 8874096 p^{3} T^{5} - 515041 p^{6} T^{6} + 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 896 T + 800494 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 966 T - 63367 T^{2} + 39606 p^{2} T^{3} + 1230564 p^{3} T^{4} + 39606 p^{5} T^{5} - 63367 p^{6} T^{6} + 966 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 52 T - 329557 T^{2} + 33998484 T^{3} - 134023260856 T^{4} + 33998484 p^{3} T^{5} - 329557 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 392 T + 895462 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 294 T - 1211911 T^{2} - 32807754 T^{3} + 1127788940964 T^{4} - 32807754 p^{3} T^{5} - 1211911 p^{6} T^{6} + 294 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 420 T + 1655734 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−12.58948648117584964655194266480, −11.99714715958760638046309173637, −11.92925190716177768207165890286, −11.46201711689485749598524467180, −11.43864920570591855030025090275, −10.45971564049600003129484850174, −10.41204732926217839515874382774, −9.946176194892261544745350455492, −9.759168889496545663055424274214, −9.718971425808629794624026418556, −8.987292020894827369800012545155, −8.499630381252146260939173600884, −8.106745025723087270917930925957, −7.62411085058911186276509070122, −7.40542310372798220637208654890, −7.13620111543091371005115488851, −6.46513528920593768258583569211, −5.68504951299888844705207226744, −5.25779715443103296975800903626, −5.13517691840469988653143518796, −4.96551126703948252761080937000, −3.50811295432978904342441911189, −3.32278816969897170810648176798, −2.03459617064978623957679647539, −1.47758963390656241625822181562,
1.47758963390656241625822181562, 2.03459617064978623957679647539, 3.32278816969897170810648176798, 3.50811295432978904342441911189, 4.96551126703948252761080937000, 5.13517691840469988653143518796, 5.25779715443103296975800903626, 5.68504951299888844705207226744, 6.46513528920593768258583569211, 7.13620111543091371005115488851, 7.40542310372798220637208654890, 7.62411085058911186276509070122, 8.106745025723087270917930925957, 8.499630381252146260939173600884, 8.987292020894827369800012545155, 9.718971425808629794624026418556, 9.759168889496545663055424274214, 9.946176194892261544745350455492, 10.41204732926217839515874382774, 10.45971564049600003129484850174, 11.43864920570591855030025090275, 11.46201711689485749598524467180, 11.92925190716177768207165890286, 11.99714715958760638046309173637, 12.58948648117584964655194266480
