L(s) = 1 | + 4-s − 4·5-s + 9-s − 10·11-s + 16·19-s − 4·20-s + 2·25-s + 14·29-s − 28·31-s + 36-s − 10·44-s − 4·45-s + 2·49-s + 40·55-s − 10·59-s − 8·61-s − 64-s − 16·71-s + 16·76-s − 4·79-s − 64·95-s − 10·99-s + 2·100-s + 4·101-s + 48·109-s + 14·116-s + 47·121-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1.78·5-s + 1/3·9-s − 3.01·11-s + 3.67·19-s − 0.894·20-s + 2/5·25-s + 2.59·29-s − 5.02·31-s + 1/6·36-s − 1.50·44-s − 0.596·45-s + 2/7·49-s + 5.39·55-s − 1.30·59-s − 1.02·61-s − 1/8·64-s − 1.89·71-s + 1.83·76-s − 0.450·79-s − 6.56·95-s − 1.00·99-s + 1/5·100-s + 0.398·101-s + 4.59·109-s + 1.29·116-s + 4.27·121-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
\(L(1)\) |
\(\approx\) |
\(0.5811342231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5811342231\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.074438177838774821909957434788, −7.954398634600786693710233690185, −7.67679244320115573065542070755, −7.41875507973036121836403094390, −7.37394427777795772660664961049, −7.20918304551257752286449801325, −6.83217662294791915136496288166, −6.73532333170398128983585171558, −5.85876874669041225382671739283, −5.79978148466395964757069754595, −5.66652588577871367726047655829, −5.43284851806162721190207906954, −5.16407868492461975782949194343, −4.63200287293235525627452814158, −4.57312119247027176292335823121, −4.45635579112063896931716342066, −3.63170623858383720726006229690, −3.48468269755921177814832383661, −3.32097360311064545452806388245, −2.97265521241829789945219663182, −2.81966184613366577120819018371, −2.09673229041402285743385754765, −1.86470832754220607218138244917, −1.12932982314036409475220127700, −0.32046737557729335885413061771,
0.32046737557729335885413061771, 1.12932982314036409475220127700, 1.86470832754220607218138244917, 2.09673229041402285743385754765, 2.81966184613366577120819018371, 2.97265521241829789945219663182, 3.32097360311064545452806388245, 3.48468269755921177814832383661, 3.63170623858383720726006229690, 4.45635579112063896931716342066, 4.57312119247027176292335823121, 4.63200287293235525627452814158, 5.16407868492461975782949194343, 5.43284851806162721190207906954, 5.66652588577871367726047655829, 5.79978148466395964757069754595, 5.85876874669041225382671739283, 6.73532333170398128983585171558, 6.83217662294791915136496288166, 7.20918304551257752286449801325, 7.37394427777795772660664961049, 7.41875507973036121836403094390, 7.67679244320115573065542070755, 7.954398634600786693710233690185, 8.074438177838774821909957434788