| L(s) = 1 | + 4-s + 4·5-s + 9-s − 6·11-s + 4·20-s + 2·25-s − 14·29-s + 4·31-s + 36-s − 6·44-s + 4·45-s − 14·49-s − 24·55-s − 22·59-s − 64-s − 8·71-s − 4·79-s − 8·89-s − 6·99-s + 2·100-s + 28·101-s + 64·109-s − 14·116-s + 31·121-s + 4·124-s − 28·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s + 1/3·9-s − 1.80·11-s + 0.894·20-s + 2/5·25-s − 2.59·29-s + 0.718·31-s + 1/6·36-s − 0.904·44-s + 0.596·45-s − 2·49-s − 3.23·55-s − 2.86·59-s − 1/8·64-s − 0.949·71-s − 0.450·79-s − 0.847·89-s − 0.603·99-s + 1/5·100-s + 2.78·101-s + 6.13·109-s − 1.29·116-s + 2.81·121-s + 0.359·124-s − 2.50·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253011786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253011786\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 - 22 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 + 27 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 178 T^{2} + 22275 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.308745854675354611959574869291, −7.81252339959170431792992238353, −7.52968644524750381490817055345, −7.49469052842598502006210800059, −7.33746365557553125324320942697, −7.11840521164862702860675046279, −6.52115377836540307523463450004, −6.32044922050739498007105159039, −6.11866823091013366769452632513, −5.95030175853905733310980143311, −5.73619607287467534759947723992, −5.40095545594166892986445322971, −5.27957983684157521839533100709, −4.70081728248280564190534106947, −4.59600379550292898199093262875, −4.52530387999619851243271388278, −3.81633843893756395159065008809, −3.43246629461652458233756253495, −3.09661721052871470526448639099, −2.99300254036233513121987740804, −2.44295096399459258998585735692, −1.87860998897190478289866382792, −1.86049703780094644396014637768, −1.73342209136597312468964981183, −0.52346946873224122951136894530,
0.52346946873224122951136894530, 1.73342209136597312468964981183, 1.86049703780094644396014637768, 1.87860998897190478289866382792, 2.44295096399459258998585735692, 2.99300254036233513121987740804, 3.09661721052871470526448639099, 3.43246629461652458233756253495, 3.81633843893756395159065008809, 4.52530387999619851243271388278, 4.59600379550292898199093262875, 4.70081728248280564190534106947, 5.27957983684157521839533100709, 5.40095545594166892986445322971, 5.73619607287467534759947723992, 5.95030175853905733310980143311, 6.11866823091013366769452632513, 6.32044922050739498007105159039, 6.52115377836540307523463450004, 7.11840521164862702860675046279, 7.33746365557553125324320942697, 7.49469052842598502006210800059, 7.52968644524750381490817055345, 7.81252339959170431792992238353, 8.308745854675354611959574869291
