| L(s) = 1 | + 7·3-s − 133·5-s − 72·7-s − 51·9-s + 705·11-s − 1.34e3·13-s − 931·15-s + 2.78e3·17-s + 722·19-s − 504·21-s + 2.71e3·23-s + 8.12e3·25-s + 644·27-s − 7.77e3·29-s − 7.13e3·31-s + 4.93e3·33-s + 9.57e3·35-s − 6.24e3·37-s − 9.38e3·39-s − 4.17e3·41-s − 2.53e4·43-s + 6.78e3·45-s − 1.17e4·47-s − 2.10e4·49-s + 1.94e4·51-s − 2.91e4·53-s − 9.37e4·55-s + ⋯ |
| L(s) = 1 | + 0.449·3-s − 2.37·5-s − 0.555·7-s − 0.209·9-s + 1.75·11-s − 2.20·13-s − 1.06·15-s + 2.33·17-s + 0.458·19-s − 0.249·21-s + 1.06·23-s + 2.59·25-s + 0.170·27-s − 1.71·29-s − 1.33·31-s + 0.788·33-s + 1.32·35-s − 0.750·37-s − 0.988·39-s − 0.387·41-s − 2.09·43-s + 0.499·45-s − 0.774·47-s − 1.25·49-s + 1.04·51-s − 1.42·53-s − 4.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 7 T + 100 T^{2} - 7 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 133 T + 9566 T^{2} + 133 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 72 T + 26237 T^{2} + 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 705 T + 438880 T^{2} - 705 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1341 T + 1154942 T^{2} + 1341 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2784 T + 4699321 T^{2} - 2784 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2713 T + 11414870 T^{2} - 2713 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7775 T + 54717140 T^{2} + 7775 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7132 T + 35395230 T^{2} + 7132 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6248 T + 110667702 T^{2} + 6248 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4174 T + 178762274 T^{2} + 4174 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 25357 T + 397354794 T^{2} + 25357 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11727 T + 493067062 T^{2} + 11727 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 29133 T + 973906414 T^{2} + 29133 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 64515 T + 2470044400 T^{2} - 64515 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 40939 T + 1409015376 T^{2} + 40939 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 19039 T + 2619663906 T^{2} + 19039 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 70236 T + 3438148114 T^{2} - 70236 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 67058 T + 4701943299 T^{2} - 67058 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 32850 T + 6247247246 T^{2} - 32850 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 71534 T + 8544351758 T^{2} + 71534 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 87268 T + 9132109106 T^{2} + 87268 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 62458 T + 18006879378 T^{2} + 62458 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.78961479050163505425040722176, −9.933114045744832465960600380736, −9.543206377029405252773762516591, −9.479082934766200943925858364993, −8.573673304086815491825260781313, −8.234336730302614550850017559191, −7.54000688315256033377952032150, −7.50382125086010780185244560034, −6.94477639805464491508390299994, −6.52165936658402666134977796882, −5.25289191792633019445191960304, −5.23572575041254694686393114256, −4.34012810750121609853593449231, −3.67433871803768230327350662233, −3.27202617017364487653970107527, −3.22748587929676495102266399562, −1.86780022985127138262132988996, −1.08229411507150199079817303957, 0, 0,
1.08229411507150199079817303957, 1.86780022985127138262132988996, 3.22748587929676495102266399562, 3.27202617017364487653970107527, 3.67433871803768230327350662233, 4.34012810750121609853593449231, 5.23572575041254694686393114256, 5.25289191792633019445191960304, 6.52165936658402666134977796882, 6.94477639805464491508390299994, 7.50382125086010780185244560034, 7.54000688315256033377952032150, 8.234336730302614550850017559191, 8.573673304086815491825260781313, 9.479082934766200943925858364993, 9.543206377029405252773762516591, 9.933114045744832465960600380736, 10.78961479050163505425040722176
