| L(s) = 1 | + 2·7-s + 2·13-s − 4·19-s + 8·31-s + 14·37-s + 2·43-s − 15·49-s + 14·61-s + 20·67-s + 20·73-s − 16·79-s + 4·91-s + 32·97-s + 14·103-s − 22·109-s − 11·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 39·169-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.554·13-s − 0.917·19-s + 1.43·31-s + 2.30·37-s + 0.304·43-s − 2.14·49-s + 1.79·61-s + 2.44·67-s + 2.34·73-s − 1.80·79-s + 0.419·91-s + 3.24·97-s + 1.37·103-s − 2.10·109-s − 121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.475562540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.475562540\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + p T^{2} + 225 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 35 T^{2} + 837 T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 59 T^{2} + 1881 T^{4} + 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 119 T^{2} + 6477 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - T + 81 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 179 T^{2} + 13581 T^{4} + 179 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 167 T^{2} + 12753 T^{4} + 167 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 110 T^{2} + 4707 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 176 T^{2} + 16782 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.58253710562728000675031460990, −5.11939822387148126084013133108, −5.02476158513407042053153331941, −4.97689017198815235249806607235, −4.91586418714375076394766861414, −4.42047815639911131566349084101, −4.33683327533803074326801817723, −4.29995306137915280329030198040, −4.17083961694389233292343297572, −3.60982681464726238118186452908, −3.59041061018980204964332271305, −3.49787981923287348867538196522, −3.41828105837087049908956265758, −2.76517811742840431543534865094, −2.71111766245530919875862693057, −2.59467552708757480200903545535, −2.49597360375491267286784153073, −2.01309524670960504334840675853, −1.86728320476343154805638531494, −1.74894485648376808100540525275, −1.43523254633266185264072622850, −1.07139471488209987966550203072, −0.72218947432561257461091312333, −0.67963789628536850746450384065, −0.39526157380419416790020628603,
0.39526157380419416790020628603, 0.67963789628536850746450384065, 0.72218947432561257461091312333, 1.07139471488209987966550203072, 1.43523254633266185264072622850, 1.74894485648376808100540525275, 1.86728320476343154805638531494, 2.01309524670960504334840675853, 2.49597360375491267286784153073, 2.59467552708757480200903545535, 2.71111766245530919875862693057, 2.76517811742840431543534865094, 3.41828105837087049908956265758, 3.49787981923287348867538196522, 3.59041061018980204964332271305, 3.60982681464726238118186452908, 4.17083961694389233292343297572, 4.29995306137915280329030198040, 4.33683327533803074326801817723, 4.42047815639911131566349084101, 4.91586418714375076394766861414, 4.97689017198815235249806607235, 5.02476158513407042053153331941, 5.11939822387148126084013133108, 5.58253710562728000675031460990
