| L(s) = 1 | − 7.86e3·5-s − 9.10e4·7-s + 1.59e5·11-s + 1.05e6·13-s − 1.43e6·17-s + 2.18e7·19-s + 3.58e7·23-s + 1.30e7·25-s + 2.28e8·29-s − 6.47e7·31-s + 7.16e8·35-s + 7.55e7·37-s − 1.20e9·41-s + 4.55e7·43-s − 1.22e9·47-s + 3.05e9·49-s + 3.80e9·53-s − 1.25e9·55-s − 6.01e9·59-s + 9.78e9·61-s − 8.26e9·65-s − 1.47e10·67-s + 4.31e9·71-s + 1.10e10·73-s − 1.44e10·77-s − 5.19e10·79-s − 1.08e11·83-s + ⋯ |
| L(s) = 1 | − 1.12·5-s − 2.04·7-s + 0.297·11-s + 0.784·13-s − 0.244·17-s + 2.02·19-s + 1.16·23-s + 0.267·25-s + 2.07·29-s − 0.406·31-s + 2.30·35-s + 0.179·37-s − 1.61·41-s + 0.0472·43-s − 0.781·47-s + 1.54·49-s + 1.25·53-s − 0.335·55-s − 1.09·59-s + 1.48·61-s − 0.883·65-s − 1.33·67-s + 0.284·71-s + 0.624·73-s − 0.609·77-s − 1.89·79-s − 3.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | $D_{4}$ | \( 1 + 7868 T + 9768358 p T^{2} + 7868 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 13008 p T + 106936526 p^{2} T^{2} + 13008 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 159080 T + 362536063286 T^{2} - 159080 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1050476 T + 3458956762062 T^{2} - 1050476 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1430884 T + 16748384809766 T^{2} + 1430884 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 21866600 T + 343123710088422 T^{2} - 21866600 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 35806736 T + 1952928132896462 T^{2} - 35806736 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 228827700 T + 35907977791027054 T^{2} - 228827700 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 64722112 T + 51498184379920062 T^{2} + 64722112 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 75558780 T + 354292341022528382 T^{2} - 75558780 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1201214196 T + 1274442257818453270 T^{2} + 1201214196 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 45519832 T + 42144714696540642 p T^{2} - 45519832 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1229079264 T + 5024390553242310430 T^{2} + 1229079264 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3808549924 T + 16648234587904299038 T^{2} - 3808549924 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6012926584 T + 61066799326040273366 T^{2} + 6012926584 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} - 9789792908 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14703095224 T + 95414710392374126214 T^{2} + 14703095224 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} - 4319991088 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - 11055639476 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} + 51957623264 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} + 108227975912 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} + 71188291860 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + 1699807676 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−10.76094993475478022857014948951, −10.04019832402482948253316795526, −9.857139489102444227316630949654, −9.272775996685600829397366868849, −8.522531480160321654721144212799, −8.449406016384235792433944074558, −7.29150567499915328175918780679, −7.22589918698684590319514309506, −6.54479712157431236802494796855, −6.15029614293764031107328411814, −5.36681911830503202254693624194, −4.77564686000523987077298157145, −3.90723600983348152413423128292, −3.58595530486021831993528179316, −2.90496118383092103394511157418, −2.83719969140385370720182510424, −1.27047461205700032548488511476, −1.06629846719119338761458902232, 0, 0,
1.06629846719119338761458902232, 1.27047461205700032548488511476, 2.83719969140385370720182510424, 2.90496118383092103394511157418, 3.58595530486021831993528179316, 3.90723600983348152413423128292, 4.77564686000523987077298157145, 5.36681911830503202254693624194, 6.15029614293764031107328411814, 6.54479712157431236802494796855, 7.22589918698684590319514309506, 7.29150567499915328175918780679, 8.449406016384235792433944074558, 8.522531480160321654721144212799, 9.272775996685600829397366868849, 9.857139489102444227316630949654, 10.04019832402482948253316795526, 10.76094993475478022857014948951
