| L(s) = 1 | − 2·3-s + 28·9-s + 42·11-s + 92·13-s + 4·17-s + 36·19-s + 258·23-s + 474·25-s − 52·27-s + 270·29-s − 84·33-s − 390·37-s − 184·39-s + 648·41-s + 238·43-s + 49·49-s − 8·51-s − 44·53-s − 72·57-s + 1.37e3·59-s + 1.70e3·61-s − 534·67-s − 516·69-s + 1.51e3·71-s − 948·75-s − 1.69e3·79-s + 725·81-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.03·9-s + 1.15·11-s + 1.96·13-s + 0.0570·17-s + 0.434·19-s + 2.33·23-s + 3.79·25-s − 0.370·27-s + 1.72·29-s − 0.443·33-s − 1.73·37-s − 0.755·39-s + 2.46·41-s + 0.844·43-s + 1/7·49-s − 0.0219·51-s − 0.114·53-s − 0.167·57-s + 3.03·59-s + 3.57·61-s − 0.973·67-s − 0.900·69-s + 2.52·71-s − 1.45·75-s − 2.40·79-s + 0.994·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.61187011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.61187011\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 92 T + 399 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - 8 p T^{2} - 52 T^{3} - 53 T^{4} - 52 p^{3} T^{5} - 8 p^{7} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 474 T^{2} + 87371 T^{4} - 474 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 42 T + 1876 T^{2} - 54096 T^{3} + 670011 T^{4} - 54096 p^{3} T^{5} + 1876 p^{6} T^{6} - 42 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 3187 T^{2} + 26492 T^{3} - 13956968 T^{4} + 26492 p^{3} T^{5} - 3187 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 36 T + 10894 T^{2} - 376632 T^{3} + 65370651 T^{4} - 376632 p^{3} T^{5} + 10894 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 258 T + 25952 T^{2} - 4199724 T^{3} + 691414467 T^{4} - 4199724 p^{3} T^{5} + 25952 p^{6} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 270 T + 7097 T^{2} - 4596750 T^{3} + 1957253388 T^{4} - 4596750 p^{3} T^{5} + 7097 p^{6} T^{6} - 270 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68548 T^{2} + 2713470810 T^{4} - 68548 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 390 T + 97081 T^{2} + 18088590 T^{3} + 2153577852 T^{4} + 18088590 p^{3} T^{5} + 97081 p^{6} T^{6} + 390 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 648 T + 311281 T^{2} - 111010824 T^{3} + 34244774256 T^{4} - 111010824 p^{3} T^{5} + 311281 p^{6} T^{6} - 648 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 238 T - 65048 T^{2} + 8882636 T^{3} + 3983985307 T^{4} + 8882636 p^{3} T^{5} - 65048 p^{6} T^{6} - 238 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 286140 T^{2} + 41271694406 T^{4} - 286140 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 22 T + 230375 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1374 T + 1132348 T^{2} - 691198944 T^{3} + 340128167163 T^{4} - 691198944 p^{3} T^{5} + 1132348 p^{6} T^{6} - 1374 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1704 T + 1724425 T^{2} - 20258856 p T^{3} + 182721144 p^{2} T^{4} - 20258856 p^{4} T^{5} + 1724425 p^{6} T^{6} - 1704 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 534 T + 707572 T^{2} + 327085680 T^{3} + 313310492907 T^{4} + 327085680 p^{3} T^{5} + 707572 p^{6} T^{6} + 534 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1512 T + 1322638 T^{2} - 847612080 T^{3} + 458906225907 T^{4} - 847612080 p^{3} T^{5} + 1322638 p^{6} T^{6} - 1512 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 617522 T^{2} + 295619638611 T^{4} - 617522 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 846 T + 999332 T^{2} + 846 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1459524 T^{2} + 1067068480682 T^{4} - 1459524 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1212 T + 2000682 T^{2} - 1831373208 T^{3} + 2131429119107 T^{4} - 1831373208 p^{3} T^{5} + 2000682 p^{6} T^{6} - 1212 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4188 T + 8869210 T^{2} - 12659327256 T^{3} + 13640013339219 T^{4} - 12659327256 p^{3} T^{5} + 8869210 p^{6} T^{6} - 4188 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.85843013549582434364955899392, −7.27085820905553820187734439574, −7.12324705928028812930853525135, −7.07841234499293759943762394706, −7.07402937043215255414547076978, −6.49736990194298076218406704229, −6.31345017241367800034230544641, −6.11594742105160518147085177714, −6.02921925295908722058997180967, −5.33636273183518164393207995860, −4.99405879796429930254349325325, −4.96651569112661969024437762634, −4.91759677266133637061715108326, −4.42058549313857432153459825688, −3.84971176891717250769568707240, −3.71942127621318531151715013755, −3.68741403303621503335126859164, −3.08283629256766333719312210750, −2.88686720456403833147051627177, −2.28338624115042772595736180972, −2.17814443374241827980261741305, −1.13479019968348155188081280724, −1.04838145036201523742670508238, −0.897730319455105337354250839235, −0.867716188682110635396489647224,
0.867716188682110635396489647224, 0.897730319455105337354250839235, 1.04838145036201523742670508238, 1.13479019968348155188081280724, 2.17814443374241827980261741305, 2.28338624115042772595736180972, 2.88686720456403833147051627177, 3.08283629256766333719312210750, 3.68741403303621503335126859164, 3.71942127621318531151715013755, 3.84971176891717250769568707240, 4.42058549313857432153459825688, 4.91759677266133637061715108326, 4.96651569112661969024437762634, 4.99405879796429930254349325325, 5.33636273183518164393207995860, 6.02921925295908722058997180967, 6.11594742105160518147085177714, 6.31345017241367800034230544641, 6.49736990194298076218406704229, 7.07402937043215255414547076978, 7.07841234499293759943762394706, 7.12324705928028812930853525135, 7.27085820905553820187734439574, 7.85843013549582434364955899392
