| L(s) = 1 | + 3-s − 10·5-s − 8·9-s + 47·11-s − 218·13-s − 10·15-s + 121·17-s − 156·19-s + 152·23-s + 25·25-s − 79·27-s − 30·29-s − 142·31-s + 47·33-s − 46·37-s − 218·39-s + 620·41-s − 1.89e3·43-s + 80·45-s + 131·47-s + 121·51-s − 344·53-s − 470·55-s − 156·57-s − 976·59-s + 898·61-s + 2.18e3·65-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.894·5-s − 0.296·9-s + 1.28·11-s − 4.65·13-s − 0.172·15-s + 1.72·17-s − 1.88·19-s + 1.37·23-s + 1/5·25-s − 0.563·27-s − 0.192·29-s − 0.822·31-s + 0.247·33-s − 0.204·37-s − 0.895·39-s + 2.36·41-s − 6.70·43-s + 0.265·45-s + 0.406·47-s + 0.332·51-s − 0.891·53-s − 1.15·55-s − 0.362·57-s − 2.15·59-s + 1.88·61-s + 4.15·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4630536852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4630536852\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - T + p^{2} T^{2} + 62 T^{3} - 692 T^{4} + 62 p^{3} T^{5} + p^{8} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 47 T + 551 T^{2} + 47188 T^{3} - 1962776 T^{4} + 47188 p^{3} T^{5} + 551 p^{6} T^{6} - 47 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 109 T + 6804 T^{2} + 109 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 121 T + 2711 T^{2} - 254584 T^{3} + 46256098 T^{4} - 254584 p^{3} T^{5} + 2711 p^{6} T^{6} - 121 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 156 T + 8518 T^{2} + 327600 T^{3} + 36242619 T^{4} + 327600 p^{3} T^{5} + 8518 p^{6} T^{6} + 156 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 152 T + 8930 T^{2} + 1544320 T^{3} - 228239981 T^{4} + 1544320 p^{3} T^{5} + 8930 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 15 T + 45784 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 142 T - 32258 T^{2} - 1016720 T^{3} + 1259856679 T^{4} - 1016720 p^{3} T^{5} - 32258 p^{6} T^{6} + 142 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 46 T + 10090 T^{2} - 5026880 T^{3} - 2609323481 T^{4} - 5026880 p^{3} T^{5} + 10090 p^{6} T^{6} + 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 310 T + 155642 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 22 p T + 382494 T^{2} + 22 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 131 T - 194713 T^{2} - 553868 T^{3} + 32329670044 T^{4} - 553868 p^{3} T^{5} - 194713 p^{6} T^{6} - 131 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 344 T + 78842 T^{2} - 88841440 T^{3} - 38222093765 T^{4} - 88841440 p^{3} T^{5} + 78842 p^{6} T^{6} + 344 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 976 T + 312638 T^{2} + 223679680 T^{3} + 171701003899 T^{4} + 223679680 p^{3} T^{5} + 312638 p^{6} T^{6} + 976 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 898 T + 153082 T^{2} - 179025280 T^{3} + 192270874999 T^{4} - 179025280 p^{3} T^{5} + 153082 p^{6} T^{6} - 898 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 478 T - 358202 T^{2} - 7093520 T^{3} + 185022547135 T^{4} - 7093520 p^{3} T^{5} - 358202 p^{6} T^{6} + 478 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1548 T + 1313902 T^{2} - 1548 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 530 T - 411734 T^{2} + 45262000 T^{3} + 219132756367 T^{4} + 45262000 p^{3} T^{5} - 411734 p^{6} T^{6} - 530 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 623 T - 579881 T^{2} - 11256364 T^{3} + 502593170548 T^{4} - 11256364 p^{3} T^{5} - 579881 p^{6} T^{6} + 623 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 782 T + 487454 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 484 T - 946402 T^{2} + 110971520 T^{3} + 731828803939 T^{4} + 110971520 p^{3} T^{5} - 946402 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2253 T + 3091298 T^{2} + 2253 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.82682460338368669494729323699, −6.81453703999685807564014466794, −6.34702814494538618853867641787, −6.21966004557061422586698425503, −5.68436131946498859580862227648, −5.63520054945527384774703115070, −5.28695842649607997428027834914, −4.98843725712536376683915223157, −4.79599363842569539327044152879, −4.77456519383529616836890037342, −4.69175961515894986738561787485, −4.10069245089345915629636536526, −3.84494475356405080180991936326, −3.70025858533573939051012477623, −3.30050066200235600314000423532, −3.25517451893696998585042722525, −2.79995645183109723455216065014, −2.45398790176432126155970646475, −2.40383022251562405771879213445, −1.96935584670783274553437949283, −1.66326943898557902461262206221, −1.39220352487155010589841223581, −0.864769745519326159785038487511, −0.26661701003406079394800590148, −0.19406698051983882136818282344,
0.19406698051983882136818282344, 0.26661701003406079394800590148, 0.864769745519326159785038487511, 1.39220352487155010589841223581, 1.66326943898557902461262206221, 1.96935584670783274553437949283, 2.40383022251562405771879213445, 2.45398790176432126155970646475, 2.79995645183109723455216065014, 3.25517451893696998585042722525, 3.30050066200235600314000423532, 3.70025858533573939051012477623, 3.84494475356405080180991936326, 4.10069245089345915629636536526, 4.69175961515894986738561787485, 4.77456519383529616836890037342, 4.79599363842569539327044152879, 4.98843725712536376683915223157, 5.28695842649607997428027834914, 5.63520054945527384774703115070, 5.68436131946498859580862227648, 6.21966004557061422586698425503, 6.34702814494538618853867641787, 6.81453703999685807564014466794, 6.82682460338368669494729323699
