| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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
−9.687410620915503300317373515730, −9.335655221331768284686269325540, −8.799735096233715421593201779089, −8.660800869672724941405579508582, −7.80057109143446267093607103784, −7.48264665531727623612628923696, −6.80698350239509064300589757597, −6.65684423327695195889800012004, −5.90603883611409501387287717394, −5.89627737454797210518140827787, −4.61009627140831939916631594324, −4.41670488934528737094281125418, −3.96330802792625364909261285943, −3.51951397263116010468618453912, −2.74340745285050286751272801716, −2.30385875513540273361028697756, −1.36746496809401392238964344706, −1.08502948524480030302564362342, 0, 0,
1.08502948524480030302564362342, 1.36746496809401392238964344706, 2.30385875513540273361028697756, 2.74340745285050286751272801716, 3.51951397263116010468618453912, 3.96330802792625364909261285943, 4.41670488934528737094281125418, 4.61009627140831939916631594324, 5.89627737454797210518140827787, 5.90603883611409501387287717394, 6.65684423327695195889800012004, 6.80698350239509064300589757597, 7.48264665531727623612628923696, 7.80057109143446267093607103784, 8.660800869672724941405579508582, 8.799735096233715421593201779089, 9.335655221331768284686269325540, 9.687410620915503300317373515730
