| L(s) = 1 | − 5-s − 5·7-s − 4·11-s + 14·13-s − 6·17-s − 3·19-s − 2·23-s + 4·29-s − 7·31-s + 5·35-s + 7·37-s + 16·41-s + 10·43-s + 10·47-s + 18·49-s − 8·53-s + 4·55-s + 10·59-s + 6·61-s − 14·65-s − 3·67-s − 15·73-s + 20·77-s − 79-s − 16·83-s + 6·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s − 1.20·11-s + 3.88·13-s − 1.45·17-s − 0.688·19-s − 0.417·23-s + 0.742·29-s − 1.25·31-s + 0.845·35-s + 1.15·37-s + 2.49·41-s + 1.52·43-s + 1.45·47-s + 18/7·49-s − 1.09·53-s + 0.539·55-s + 1.30·59-s + 0.768·61-s − 1.73·65-s − 0.366·67-s − 1.75·73-s + 2.27·77-s − 0.112·79-s − 1.75·83-s + 0.650·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261993982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261993982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.854841928857544441656671985006, −9.394532182480944874267188999577, −8.915855268582601702154142891928, −8.800821680521949118612347702871, −8.379815792294676150069973695893, −7.971589575672985680435574219341, −7.34947147632109602011473184340, −7.01161671093974062175757734310, −6.46413072487416308669651032960, −6.03141849305104832464210902560, −5.84890174401273701277718597407, −5.70351554180458461424543216130, −4.47587220652628025851132830824, −4.09285998962856956311189882601, −3.90886557178600735210015007608, −3.37205058030186558840454043625, −2.69372013546873908501875540809, −2.39985290215922201020112336055, −1.25171298357288026216613093534, −0.52099052843588142411081167438,
0.52099052843588142411081167438, 1.25171298357288026216613093534, 2.39985290215922201020112336055, 2.69372013546873908501875540809, 3.37205058030186558840454043625, 3.90886557178600735210015007608, 4.09285998962856956311189882601, 4.47587220652628025851132830824, 5.70351554180458461424543216130, 5.84890174401273701277718597407, 6.03141849305104832464210902560, 6.46413072487416308669651032960, 7.01161671093974062175757734310, 7.34947147632109602011473184340, 7.971589575672985680435574219341, 8.379815792294676150069973695893, 8.800821680521949118612347702871, 8.915855268582601702154142891928, 9.394532182480944874267188999577, 9.854841928857544441656671985006
