| L(s) = 1 | + 3·2-s − 2·3-s + 3·4-s − 6·6-s − 6·7-s + 7·9-s − 6·12-s − 4·13-s − 18·14-s − 2·16-s + 21·18-s + 6·19-s + 12·21-s − 12·26-s − 22·27-s − 18·28-s + 6·32-s + 21·36-s + 18·38-s + 8·39-s + 36·42-s − 12·43-s + 4·48-s + 7·49-s − 12·52-s + 12·53-s − 66·54-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 3/2·4-s − 2.44·6-s − 2.26·7-s + 7/3·9-s − 1.73·12-s − 1.10·13-s − 4.81·14-s − 1/2·16-s + 4.94·18-s + 1.37·19-s + 2.61·21-s − 2.35·26-s − 4.23·27-s − 3.40·28-s + 1.06·32-s + 7/2·36-s + 2.91·38-s + 1.28·39-s + 5.55·42-s − 1.82·43-s + 0.577·48-s + 49-s − 1.66·52-s + 1.64·53-s − 8.98·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.686598213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.686598213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^2$ | \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 15 T^{2} + 104 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 13 T^{2} - 120 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 25 T^{2} + 96 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 37 T^{2} + 528 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 1182 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 11 T^{2} - 1248 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 75 T^{2} + 3944 T^{4} + 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5990 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 30 T + 465 T^{2} - 4950 T^{3} + 41444 T^{4} - 4950 p T^{5} + 465 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 7 T^{2} - 180 T^{3} + 6264 T^{4} - 180 p T^{5} + 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 24 T + 311 T^{2} + 2856 T^{3} + 22536 T^{4} + 2856 p T^{5} + 311 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 702 T^{3} + 9500 T^{4} - 702 p T^{5} + 129 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 18734 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 411 T^{2} + 5256 T^{3} + 57128 T^{4} + 5256 p T^{5} + 411 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 191 T^{2} + 1716 T^{3} + 15696 T^{4} + 1716 p T^{5} + 191 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.371513866293306990755644036452, −8.280988521777881352649587660585, −7.62004421921352067170964512435, −7.50960220348689604534324089988, −7.06115056004437633628981360395, −7.03666798492582082078008243046, −6.76839225838831498061933969020, −6.74404776009806882788166735013, −6.22988354968613841483745737696, −5.93994624560785862265356508317, −5.68377149119135759543354849280, −5.44937184969618738718515500094, −5.26841311765211245210416347601, −5.09199326270477748059413743676, −4.66291095058155772363953286685, −4.28274589033271267660801127895, −4.07023137733667860107458978313, −3.90678491358934518309408138495, −3.73084927598201346986096791701, −3.14910051132875279422328253182, −2.87195912930888358762925714451, −2.66257937384224182243757120324, −1.88349564758820319220848150770, −1.37277825334865716856496814071, −0.44921864427236443265723462774,
0.44921864427236443265723462774, 1.37277825334865716856496814071, 1.88349564758820319220848150770, 2.66257937384224182243757120324, 2.87195912930888358762925714451, 3.14910051132875279422328253182, 3.73084927598201346986096791701, 3.90678491358934518309408138495, 4.07023137733667860107458978313, 4.28274589033271267660801127895, 4.66291095058155772363953286685, 5.09199326270477748059413743676, 5.26841311765211245210416347601, 5.44937184969618738718515500094, 5.68377149119135759543354849280, 5.93994624560785862265356508317, 6.22988354968613841483745737696, 6.74404776009806882788166735013, 6.76839225838831498061933969020, 7.03666798492582082078008243046, 7.06115056004437633628981360395, 7.50960220348689604534324089988, 7.62004421921352067170964512435, 8.280988521777881352649587660585, 8.371513866293306990755644036452
