L(s) = 1 | + 4·2-s + 3·3-s + 10·4-s + 3·5-s + 12·6-s + 20·8-s + 3·9-s + 12·10-s + 30·12-s + 10·13-s + 9·15-s + 35·16-s − 17-s + 12·18-s + 14·19-s + 30·20-s − 4·23-s + 60·24-s − 7·25-s + 40·26-s + 2·27-s − 3·29-s + 36·30-s + 3·31-s + 56·32-s − 4·34-s + 30·36-s + ⋯ |
L(s) = 1 | + 2.82·2-s + 1.73·3-s + 5·4-s + 1.34·5-s + 4.89·6-s + 7.07·8-s + 9-s + 3.79·10-s + 8.66·12-s + 2.77·13-s + 2.32·15-s + 35/4·16-s − 0.242·17-s + 2.82·18-s + 3.21·19-s + 6.70·20-s − 0.834·23-s + 12.2·24-s − 7/5·25-s + 7.84·26-s + 0.384·27-s − 0.557·29-s + 6.57·30-s + 0.538·31-s + 9.89·32-s − 0.685·34-s + 5·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(219.0417316\) |
\(L(\frac12)\) |
\(\approx\) |
\(219.0417316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 3 | $C_2 \wr S_4$ | \( 1 - p T + 2 p T^{2} - 11 T^{3} + 22 T^{4} - 11 p T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 3 T + 16 T^{2} - 37 T^{3} + 116 T^{4} - 37 p T^{5} + 16 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 24 T^{2} - 10 T^{3} + 294 T^{4} - 10 p T^{5} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 10 T + 72 T^{2} - 354 T^{3} + 1478 T^{4} - 354 p T^{5} + 72 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + T + 36 T^{2} + 115 T^{3} + 618 T^{4} + 115 p T^{5} + 36 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 14 T + 132 T^{2} - 858 T^{3} + 4310 T^{4} - 858 p T^{5} + 132 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 4 T + 52 T^{2} + 208 T^{3} + 1366 T^{4} + 208 p T^{5} + 52 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 3 T + 94 T^{2} + 259 T^{3} + 3776 T^{4} + 259 p T^{5} + 94 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 3 T + 102 T^{2} - 251 T^{3} + 4394 T^{4} - 251 p T^{5} + 102 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 14 T + 184 T^{2} + 1434 T^{3} + 10654 T^{4} + 1434 p T^{5} + 184 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - T + 112 T^{2} + 127 T^{3} + 5646 T^{4} + 127 p T^{5} + 112 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 96 T^{2} + 662 T^{3} + 4158 T^{4} + 662 p T^{5} + 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 11 T + 112 T^{2} - 723 T^{3} + 6700 T^{4} - 723 p T^{5} + 112 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 2 T + 154 T^{2} + 560 T^{3} + 11126 T^{4} + 560 p T^{5} + 154 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 15 T + 262 T^{2} - 2421 T^{3} + 24360 T^{4} - 2421 p T^{5} + 262 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4 T + 192 T^{2} - 618 T^{3} + 17954 T^{4} - 618 p T^{5} + 192 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 15 T + 340 T^{2} - 3251 T^{3} + 38214 T^{4} - 3251 p T^{5} + 340 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 228 T^{2} - 652 T^{3} + 23142 T^{4} - 652 p T^{5} + 228 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 15 T + 292 T^{2} - 3043 T^{3} + 34422 T^{4} - 3043 p T^{5} + 292 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 12 T + 286 T^{2} - 2126 T^{3} + 31506 T^{4} - 2126 p T^{5} + 286 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 35 T + 790 T^{2} - 11589 T^{3} + 128770 T^{4} - 11589 p T^{5} + 790 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5 T + 202 T^{2} + 555 T^{3} + 24202 T^{4} + 555 p T^{5} + 202 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.90753459162855648478852810175, −5.55223521703668751412350772754, −5.48357430438960828057923271195, −5.39643727433081906464103966343, −5.35423870895475935551849602676, −4.99288696164643785856577516128, −4.78328474654761547473824751904, −4.70305986427575127719742719172, −4.04034422457888366142923454987, −4.04015387013781265883521642375, −3.79111956435046487188883630060, −3.72834341262980937332797333527, −3.65844875383600319975728593394, −3.37602969748126326114151060322, −3.15070063256192697963547756046, −3.00234628216962618257188407778, −2.81792586554378225579848736896, −2.29614750774010149438880631057, −2.20623251589029973811361267006, −2.09037323240050812357864285836, −1.96976174959267585175280782737, −1.43580468721486284283281185859, −1.40027020179370862546417829754, −0.823253533004952280901787798614, −0.815284975670715478485860607215,
0.815284975670715478485860607215, 0.823253533004952280901787798614, 1.40027020179370862546417829754, 1.43580468721486284283281185859, 1.96976174959267585175280782737, 2.09037323240050812357864285836, 2.20623251589029973811361267006, 2.29614750774010149438880631057, 2.81792586554378225579848736896, 3.00234628216962618257188407778, 3.15070063256192697963547756046, 3.37602969748126326114151060322, 3.65844875383600319975728593394, 3.72834341262980937332797333527, 3.79111956435046487188883630060, 4.04015387013781265883521642375, 4.04034422457888366142923454987, 4.70305986427575127719742719172, 4.78328474654761547473824751904, 4.99288696164643785856577516128, 5.35423870895475935551849602676, 5.39643727433081906464103966343, 5.48357430438960828057923271195, 5.55223521703668751412350772754, 5.90753459162855648478852810175