L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·8-s + 5·9-s − 4·10-s − 4·11-s + 6·12-s + 8·13-s + 4·15-s + 4·17-s − 10·18-s + 6·20-s + 8·22-s + 2·23-s − 4·24-s + 25-s − 16·26-s + 10·27-s − 4·29-s − 8·30-s − 12·31-s + 6·32-s − 8·33-s − 8·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.707·8-s + 5/3·9-s − 1.26·10-s − 1.20·11-s + 1.73·12-s + 2.21·13-s + 1.03·15-s + 0.970·17-s − 2.35·18-s + 1.34·20-s + 1.70·22-s + 0.417·23-s − 0.816·24-s + 1/5·25-s − 3.13·26-s + 1.92·27-s − 0.742·29-s − 1.46·30-s − 2.15·31-s + 1.06·32-s − 1.39·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.208896680\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.208896680\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 16 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 41 T^{2} + 2 T^{3} + 1404 T^{4} + 2 p T^{5} - 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 2 T^{2} - 448 T^{3} - 3413 T^{4} - 448 p T^{5} + 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 378 T^{3} - 2436 T^{4} - 378 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22 T + 231 T^{2} + 2618 T^{3} + 27092 T^{4} + 2618 p T^{5} + 231 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 16 p T^{5} - 126 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 3264 p T^{5} + 282 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 119 T^{2} - 138 T^{3} + 12900 T^{4} - 138 p T^{5} - 119 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.966964074485525725425061489611, −8.494008524744384346938159479645, −8.394191563404487480732559468726, −8.259682323041703459512106069649, −7.61293908866609988972723830485, −7.48517025675993278541730687486, −7.33306198863298883411678033110, −7.28052637481179459480728001355, −7.20340464363351816553377519970, −6.27579742331942364270763745093, −6.10698845133830052127967041306, −5.94843241526975937592128156737, −5.67775577855541977110088462352, −5.62292745587233820661664767806, −4.82272855369867949501967489722, −4.58714474224785047569850091099, −4.05353696304865479570130336553, −4.00195259926126514333440535817, −3.57077839248044345646542693328, −2.90313420539198745803745269542, −2.66676796780779155939280597527, −2.56395133097160256416162763748, −1.71978474861932443693738345440, −1.42376474318025814830560773575, −1.13678653735719306460260474670,
1.13678653735719306460260474670, 1.42376474318025814830560773575, 1.71978474861932443693738345440, 2.56395133097160256416162763748, 2.66676796780779155939280597527, 2.90313420539198745803745269542, 3.57077839248044345646542693328, 4.00195259926126514333440535817, 4.05353696304865479570130336553, 4.58714474224785047569850091099, 4.82272855369867949501967489722, 5.62292745587233820661664767806, 5.67775577855541977110088462352, 5.94843241526975937592128156737, 6.10698845133830052127967041306, 6.27579742331942364270763745093, 7.20340464363351816553377519970, 7.28052637481179459480728001355, 7.33306198863298883411678033110, 7.48517025675993278541730687486, 7.61293908866609988972723830485, 8.259682323041703459512106069649, 8.394191563404487480732559468726, 8.494008524744384346938159479645, 8.966964074485525725425061489611