| L(s) = 1 | + 2·9-s + 12·17-s + 12·41-s − 28·49-s − 4·73-s + 9·81-s + 36·89-s + 40·97-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.91·17-s + 1.87·41-s − 4·49-s − 0.468·73-s + 81-s + 3.81·89-s + 4.06·97-s + 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.246609577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.246609577\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−6.03596950952434510151763251639, −5.95036282545968668499885879007, −5.83167351959017090698842848675, −5.56798810906579322176147291580, −5.32111319638896904877513300947, −5.02204259149807134998799818029, −4.83810488360004118419695151736, −4.83517668010488866546000748035, −4.63587454720973309898370668866, −4.30792648405243523162388824805, −4.06055528157955106569751127146, −3.74654879658612356268018537287, −3.52906941042186193753182432290, −3.37632961412445304443922075408, −3.35258774200713434548406027476, −2.90896661734940116990221698119, −2.88523792875482247434817285879, −2.43551730141395238827047347628, −2.09517920993132586701589181508, −1.78470301512211342053828682831, −1.74400638007951193498457234961, −1.37463243675774357841767654189, −0.888759081092566096352428329528, −0.76459456404511481518675418543, −0.46331887140901201831646089021,
0.46331887140901201831646089021, 0.76459456404511481518675418543, 0.888759081092566096352428329528, 1.37463243675774357841767654189, 1.74400638007951193498457234961, 1.78470301512211342053828682831, 2.09517920993132586701589181508, 2.43551730141395238827047347628, 2.88523792875482247434817285879, 2.90896661734940116990221698119, 3.35258774200713434548406027476, 3.37632961412445304443922075408, 3.52906941042186193753182432290, 3.74654879658612356268018537287, 4.06055528157955106569751127146, 4.30792648405243523162388824805, 4.63587454720973309898370668866, 4.83517668010488866546000748035, 4.83810488360004118419695151736, 5.02204259149807134998799818029, 5.32111319638896904877513300947, 5.56798810906579322176147291580, 5.83167351959017090698842848675, 5.95036282545968668499885879007, 6.03596950952434510151763251639
