| L(s) = 1 | + 180·5-s − 700·7-s − 1.08e4·11-s − 5.48e3·13-s + 1.64e4·17-s − 1.60e4·19-s − 2.43e4·23-s − 1.31e5·25-s − 1.43e5·29-s + 3.87e4·31-s − 1.26e5·35-s + 4.55e5·37-s + 7.31e5·41-s + 1.08e6·43-s − 1.56e6·47-s − 9.49e4·49-s + 2.61e6·53-s − 1.96e6·55-s − 1.73e6·59-s − 6.20e5·61-s − 9.86e5·65-s − 3.46e5·67-s + 4.24e6·71-s − 3.14e6·73-s + 7.62e6·77-s − 1.01e7·79-s − 6.44e5·83-s + ⋯ |
| L(s) = 1 | + 0.643·5-s − 0.771·7-s − 2.46·11-s − 0.691·13-s + 0.810·17-s − 0.535·19-s − 0.417·23-s − 1.68·25-s − 1.09·29-s + 0.233·31-s − 0.496·35-s + 1.47·37-s + 1.65·41-s + 2.08·43-s − 2.19·47-s − 0.115·49-s + 2.40·53-s − 1.58·55-s − 1.09·59-s − 0.349·61-s − 0.445·65-s − 0.140·67-s + 1.40·71-s − 0.946·73-s + 1.90·77-s − 2.30·79-s − 0.123·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9291825682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9291825682\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 36 p T + 32753 p T^{2} - 36 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 100 p T + 584961 T^{2} + 100 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 90 p^{2} T + 68388367 T^{2} + 90 p^{9} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5480 T + 57188634 T^{2} + 5480 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16416 T + 79007650 T^{2} - 16416 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 16024 T + 1645526562 T^{2} + 16024 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 24372 T + 2905606450 T^{2} + 24372 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 143280 T + 39622155718 T^{2} + 143280 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 38708 T + 26496803553 T^{2} - 38708 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 455620 T + 173526750366 T^{2} - 455620 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 731880 T + 472932732862 T^{2} - 731880 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1088840 T + 751093452114 T^{2} - 1088840 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1561500 T + 1424242194466 T^{2} + 1561500 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2610468 T + 3933113324245 T^{2} - 2610468 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1731960 T + 4329510506038 T^{2} + 1731960 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 620192 T + 6303036139818 T^{2} + 620192 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 346600 T + 12137122138146 T^{2} + 346600 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4242240 T + 14648438075182 T^{2} - 4242240 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3145190 T + 18855097518219 T^{2} + 3145190 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10110616 T + 57627345888222 T^{2} + 10110616 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 644202 T + 52577539308895 T^{2} + 644202 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6021000 T + 94843763242558 T^{2} - 6021000 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4098670 T + 5732921354451 T^{2} - 4098670 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.06365132486308642820570224198, −9.882268256790260996259945393221, −9.417123096991721609125370998457, −9.076850817805680662386732825379, −8.086194118367255530648913301876, −7.968819684341470279197100800839, −7.54553113936633410162087873420, −7.16953368899714330232328109379, −6.23101094600210386048578129155, −5.99695619877888340440756413268, −5.37923727152591217155249142911, −5.37359462189576261154947106116, −4.23106771468521935287891097243, −4.17016077633502235945807563156, −3.01605334548584829110600629936, −2.84276112405619116972240309816, −2.20970507292212183829634232154, −1.84601531777269668974526213361, −0.75935269517481539412687429908, −0.24401785642753182930883691537,
0.24401785642753182930883691537, 0.75935269517481539412687429908, 1.84601531777269668974526213361, 2.20970507292212183829634232154, 2.84276112405619116972240309816, 3.01605334548584829110600629936, 4.17016077633502235945807563156, 4.23106771468521935287891097243, 5.37359462189576261154947106116, 5.37923727152591217155249142911, 5.99695619877888340440756413268, 6.23101094600210386048578129155, 7.16953368899714330232328109379, 7.54553113936633410162087873420, 7.968819684341470279197100800839, 8.086194118367255530648913301876, 9.076850817805680662386732825379, 9.417123096991721609125370998457, 9.882268256790260996259945393221, 10.06365132486308642820570224198
