| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 2·8-s − 8·9-s − 4·10-s + 8·11-s − 14·13-s + 6·17-s − 16·18-s + 24·19-s − 6·20-s + 16·22-s + 3·25-s − 28·26-s − 14·29-s − 8·31-s − 6·32-s + 12·34-s − 24·36-s + 2·37-s + 48·38-s − 4·40-s − 6·41-s − 4·43-s + 24·44-s + 16·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.707·8-s − 8/3·9-s − 1.26·10-s + 2.41·11-s − 3.88·13-s + 1.45·17-s − 3.77·18-s + 5.50·19-s − 1.34·20-s + 3.41·22-s + 3/5·25-s − 5.49·26-s − 2.59·29-s − 1.43·31-s − 1.06·32-s + 2.05·34-s − 4·36-s + 0.328·37-s + 7.78·38-s − 0.632·40-s − 0.937·41-s − 0.609·43-s + 3.61·44-s + 2.38·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.005188026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.005188026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 14 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 12 T^{2} + 112 T^{3} + 2831 T^{4} + 112 p T^{5} - 12 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 47 T^{2} + 6 T^{3} + 3732 T^{4} + 6 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 112 T^{3} + 1395 T^{4} - 112 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 216 p T^{5} + 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 10 T - 39 T^{2} + 70 T^{3} + 6692 T^{4} + 70 p T^{5} - 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T - 32 T^{2} + 216 T^{3} + 13359 T^{4} + 216 p T^{5} - 32 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} - 896 T^{3} - 9261 T^{4} - 896 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T + 6 T^{2} - 896 T^{3} + 28259 T^{4} - 896 p T^{5} + 6 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.50030395644895715263044612988, −7.20816186267345972466884967230, −7.19505016904438601886433281364, −6.93023891381082877706877621861, −6.86209090588049083146662626683, −6.52821998022063435175605814625, −5.89272355207702351268705846897, −5.65163846336988181151201707159, −5.57868776707185767015735344749, −5.46634208706761620084945069477, −5.15256136406514950223557928332, −5.11977815984498563838506659718, −5.06041902576579618316353196856, −4.28260351428739860819926292435, −3.92971702860887586271467173199, −3.89654188030203016944750728332, −3.35898316871274699661793414253, −3.28979579353599899762531540391, −3.08144851496410762630810795252, −2.99561933797327250604685367889, −2.39559176952680047883823260014, −2.19084714748773218598367160122, −1.62759586374102032425066402327, −1.03895594307536833891020496875, −0.41905424541474725897567591108,
0.41905424541474725897567591108, 1.03895594307536833891020496875, 1.62759586374102032425066402327, 2.19084714748773218598367160122, 2.39559176952680047883823260014, 2.99561933797327250604685367889, 3.08144851496410762630810795252, 3.28979579353599899762531540391, 3.35898316871274699661793414253, 3.89654188030203016944750728332, 3.92971702860887586271467173199, 4.28260351428739860819926292435, 5.06041902576579618316353196856, 5.11977815984498563838506659718, 5.15256136406514950223557928332, 5.46634208706761620084945069477, 5.57868776707185767015735344749, 5.65163846336988181151201707159, 5.89272355207702351268705846897, 6.52821998022063435175605814625, 6.86209090588049083146662626683, 6.93023891381082877706877621861, 7.19505016904438601886433281364, 7.20816186267345972466884967230, 7.50030395644895715263044612988
