| L(s) = 1 | + 3-s + 4·5-s + 2·11-s + 12·13-s + 4·15-s − 4·17-s + 4·19-s + 2·23-s + 5·25-s − 27-s − 4·29-s + 2·33-s − 2·37-s + 12·39-s + 8·43-s − 12·47-s − 4·51-s + 6·53-s + 8·55-s + 4·57-s + 8·59-s + 6·61-s + 48·65-s − 8·67-s + 2·69-s − 28·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.603·11-s + 3.32·13-s + 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.417·23-s + 25-s − 0.192·27-s − 0.742·29-s + 0.348·33-s − 0.328·37-s + 1.92·39-s + 1.21·43-s − 1.75·47-s − 0.560·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.04·59-s + 0.768·61-s + 5.95·65-s − 0.977·67-s + 0.240·69-s − 3.32·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.484678582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.484678582\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.027555529438363187943283828927, −8.956046840235688403531788546853, −8.482407516619999501636047128059, −8.376111132625193368952849482231, −7.46728862802796905732601872059, −7.39982098739445310535291862525, −6.66882391126197595563885479017, −6.34511148292552063414078427128, −6.08400937354671464385571117165, −5.66918592988547226079585033228, −5.59165735081104321113295012941, −4.75917910687312562857920022937, −4.33448063768900240479575768496, −3.76178850314619155644988262670, −3.38640234730325422353778433222, −3.09217022834732506857302488714, −2.24017521120147243938185483902, −1.87530192871213229516472357084, −1.39989156679247641154991923106, −0.913352928619763097449624431112,
0.913352928619763097449624431112, 1.39989156679247641154991923106, 1.87530192871213229516472357084, 2.24017521120147243938185483902, 3.09217022834732506857302488714, 3.38640234730325422353778433222, 3.76178850314619155644988262670, 4.33448063768900240479575768496, 4.75917910687312562857920022937, 5.59165735081104321113295012941, 5.66918592988547226079585033228, 6.08400937354671464385571117165, 6.34511148292552063414078427128, 6.66882391126197595563885479017, 7.39982098739445310535291862525, 7.46728862802796905732601872059, 8.376111132625193368952849482231, 8.482407516619999501636047128059, 8.956046840235688403531788546853, 9.027555529438363187943283828927
