| L(s) = 1 | − 6·2-s − 2·3-s + 23·4-s + 12·6-s − 22·7-s − 102·8-s + 37·9-s − 28·11-s − 46·12-s + 72·13-s + 132·14-s + 388·16-s − 76·17-s − 222·18-s − 160·19-s + 44·21-s + 168·22-s − 22·23-s + 204·24-s − 432·26-s − 106·27-s − 506·28-s − 500·29-s + 132·31-s − 1.24e3·32-s + 56·33-s + 456·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.384·3-s + 23/8·4-s + 0.816·6-s − 1.18·7-s − 4.50·8-s + 1.37·9-s − 0.767·11-s − 1.10·12-s + 1.53·13-s + 2.51·14-s + 6.06·16-s − 1.08·17-s − 2.90·18-s − 1.93·19-s + 0.457·21-s + 1.62·22-s − 0.199·23-s + 1.73·24-s − 3.25·26-s − 0.755·27-s − 3.41·28-s − 3.20·29-s + 0.764·31-s − 6.86·32-s + 0.295·33-s + 2.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3115721996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3115721996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 22 T + 51 p T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 p T + 13 T^{2} + 21 p T^{3} + 177 T^{4} + 21 p^{4} T^{5} + 13 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - 11 p T^{2} - 34 T^{3} + 532 T^{4} - 34 p^{3} T^{5} - 11 p^{7} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 28 T - 1874 T^{2} - 112 T^{3} + 4249899 T^{4} - 112 p^{3} T^{5} - 1874 p^{6} T^{6} + 28 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 36 T + 4518 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 76 T + 338 T^{2} - 333488 T^{3} - 22943213 T^{4} - 333488 p^{3} T^{5} + 338 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 160 T + 6834 T^{2} + 807680 T^{3} + 129526475 T^{4} + 807680 p^{3} T^{5} + 6834 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 22 T + 3407 T^{2} - 599654 T^{3} - 145380788 T^{4} - 599654 p^{3} T^{5} + 3407 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 250 T + 57203 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 132 T - 26514 T^{2} + 2065008 T^{3} + 523965779 T^{4} + 2065008 p^{3} T^{5} - 26514 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 416 T + 65478 T^{2} + 2609152 T^{3} + 241494107 T^{4} + 2609152 p^{3} T^{5} + 65478 p^{6} T^{6} + 416 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 106 T + 138851 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 666 T + 269853 T^{2} - 666 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 196 T - 170642 T^{2} + 276752 T^{3} + 28937567667 T^{4} + 276752 p^{3} T^{5} - 170642 p^{6} T^{6} + 196 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 952 T + 422302 T^{2} + 177308096 T^{3} + 77165754267 T^{4} + 177308096 p^{3} T^{5} + 422302 p^{6} T^{6} + 952 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 840 T + 4406 p T^{2} + 29305920 T^{3} + 11504401275 T^{4} + 29305920 p^{3} T^{5} + 4406 p^{7} T^{6} + 840 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 98 T + 203041 T^{2} - 63445102 T^{3} - 16282426916 T^{4} - 63445102 p^{3} T^{5} + 203041 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1286 T + 648343 T^{2} + 519450122 T^{3} + 423076706692 T^{4} + 519450122 p^{3} T^{5} + 648343 p^{6} T^{6} + 1286 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1064 T + 753846 T^{2} - 1064 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 172 T - 727998 T^{2} - 3517744 T^{3} + 411087581507 T^{4} - 3517744 p^{3} T^{5} - 727998 p^{6} T^{6} + 172 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1240 T + 258714 T^{2} - 363081920 T^{3} + 634365179075 T^{4} - 363081920 p^{3} T^{5} + 258714 p^{6} T^{6} - 1240 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1906 T + 2051733 T^{2} - 1906 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 650 T - 883111 T^{2} - 67812550 T^{3} + 909789389860 T^{4} - 67812550 p^{3} T^{5} - 883111 p^{6} T^{6} + 650 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 628 T + 1423942 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−9.155430338939792693972064695472, −8.651923546823838100460404959365, −8.219331503408069841046718738193, −8.214380174712989311258142045921, −7.70724009835624850480887534301, −7.58227637590329137042133587596, −7.31475374520168267770387888868, −6.88231215486654320412216756869, −6.48150195566552288247011056265, −6.41686348523769278359931107525, −6.24034928452705545816064760300, −5.94100655226129177340568679603, −5.72130335385333464035912632965, −5.12634017372541266852077381579, −4.79282411161444895706586429762, −4.18114414935220950769596066260, −3.87502284639498457921377694002, −3.50760795177842119803997980588, −3.31415683198840291245710616870, −2.60273616426828695698773556412, −2.24892041179118597165120287479, −1.83525907068060927129900948851, −1.46230159442991157565174763748, −0.44381878714096277187893656264, −0.37510614384494302072973652534,
0.37510614384494302072973652534, 0.44381878714096277187893656264, 1.46230159442991157565174763748, 1.83525907068060927129900948851, 2.24892041179118597165120287479, 2.60273616426828695698773556412, 3.31415683198840291245710616870, 3.50760795177842119803997980588, 3.87502284639498457921377694002, 4.18114414935220950769596066260, 4.79282411161444895706586429762, 5.12634017372541266852077381579, 5.72130335385333464035912632965, 5.94100655226129177340568679603, 6.24034928452705545816064760300, 6.41686348523769278359931107525, 6.48150195566552288247011056265, 6.88231215486654320412216756869, 7.31475374520168267770387888868, 7.58227637590329137042133587596, 7.70724009835624850480887534301, 8.214380174712989311258142045921, 8.219331503408069841046718738193, 8.651923546823838100460404959365, 9.155430338939792693972064695472
