| L(s) = 1 | − 2·2-s + 4-s − 4·7-s + 2·8-s − 4·13-s + 8·14-s − 4·16-s + 20·19-s + 8·26-s − 4·28-s − 4·31-s + 2·32-s − 16·37-s − 40·38-s + 18·41-s − 10·43-s − 12·47-s + 12·49-s − 4·52-s + 24·53-s − 8·56-s − 6·59-s − 16·61-s + 8·62-s + 3·64-s + 14·67-s − 24·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s + 0.707·8-s − 1.10·13-s + 2.13·14-s − 16-s + 4.58·19-s + 1.56·26-s − 0.755·28-s − 0.718·31-s + 0.353·32-s − 2.63·37-s − 6.48·38-s + 2.81·41-s − 1.52·43-s − 1.75·47-s + 12/7·49-s − 0.554·52-s + 3.29·53-s − 1.06·56-s − 0.781·59-s − 2.04·61-s + 1.01·62-s + 3/8·64-s + 1.71·67-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6366881617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6366881617\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 190 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 18 p T^{5} - 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 848 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 18 p T^{5} - 133 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 190 p T^{5} - 95 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.97554314916157853621775659179, −6.96469164746281007225269853615, −6.52854642343283407729868194926, −6.13716080164507040364570553740, −5.89015824158221458873918492438, −5.88620968383546369561951242023, −5.71106638383866389712650997756, −5.09101543743256552577129754126, −5.03821614158340195656418184784, −4.98750045357892932365784600579, −4.97328900752775700471925369182, −4.27833021072743980906953713822, −4.10127385696797997119672722961, −3.72797950645469542711784595591, −3.44911574343316387898521412932, −3.43252292856135422807364686328, −3.01050848999594023747254095874, −2.81172627278520045367829111302, −2.75242108249649347458942398009, −2.06367613286116996381539154561, −1.81132451814986329890437104502, −1.54982909476253425524237356449, −0.921685271810782544767592998191, −0.828164511139725807232619152347, −0.29394151302254968708891362939,
0.29394151302254968708891362939, 0.828164511139725807232619152347, 0.921685271810782544767592998191, 1.54982909476253425524237356449, 1.81132451814986329890437104502, 2.06367613286116996381539154561, 2.75242108249649347458942398009, 2.81172627278520045367829111302, 3.01050848999594023747254095874, 3.43252292856135422807364686328, 3.44911574343316387898521412932, 3.72797950645469542711784595591, 4.10127385696797997119672722961, 4.27833021072743980906953713822, 4.97328900752775700471925369182, 4.98750045357892932365784600579, 5.03821614158340195656418184784, 5.09101543743256552577129754126, 5.71106638383866389712650997756, 5.88620968383546369561951242023, 5.89015824158221458873918492438, 6.13716080164507040364570553740, 6.52854642343283407729868194926, 6.96469164746281007225269853615, 6.97554314916157853621775659179
