| L(s) = 1 | + 15·4-s − 224·11-s − 799·16-s − 2.84e3·19-s − 8.30e3·29-s − 1.13e4·31-s − 1.08e4·41-s − 3.36e3·44-s + 3.34e4·49-s − 5.10e4·59-s + 2.35e4·61-s − 2.73e4·64-s + 7.19e4·71-s − 4.26e4·76-s + 1.04e5·79-s − 6.77e4·89-s − 1.26e5·101-s − 1.03e5·109-s − 1.24e5·116-s − 2.84e5·121-s − 1.70e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.468·4-s − 0.558·11-s − 0.780·16-s − 1.80·19-s − 1.83·29-s − 2.12·31-s − 1.00·41-s − 0.261·44-s + 1.99·49-s − 1.90·59-s + 0.810·61-s − 0.834·64-s + 1.69·71-s − 0.846·76-s + 1.89·79-s − 0.906·89-s − 1.22·101-s − 0.831·109-s − 0.859·116-s − 1.76·121-s − 0.996·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1587494279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1587494279\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 33470 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 112 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 206090 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1921410 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1420 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2530030 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4150 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5688 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96671590 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5402 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 179654810 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 458554590 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 677983590 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 25520 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 11782 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2526326870 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 35968 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1210047410 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 52440 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3112111990 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 33870 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3286333250 T^{2} + p^{10} T^{4} \) |
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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
−11.49840841845543536633877282910, −10.89659567734179704926395564663, −10.87760989637962549432871104837, −10.38581551283570033324494198450, −9.541081905187289685520141516194, −9.202662430520918518043337004717, −8.721962844816806889443435667476, −8.142079716471310775673007806819, −7.51147645932417237789624389517, −7.15848843667040848804135127934, −6.53837769053218520559871320722, −6.05521026819151820508235066682, −5.34828858761920811791338571452, −4.93759511267029085008996712189, −3.87174737686576685806000355997, −3.79341152953553982355670377205, −2.55041444672249817541957477248, −2.17596190404951039321046793220, −1.49144389475039507538164593617, −0.10927621687906626345668425081,
0.10927621687906626345668425081, 1.49144389475039507538164593617, 2.17596190404951039321046793220, 2.55041444672249817541957477248, 3.79341152953553982355670377205, 3.87174737686576685806000355997, 4.93759511267029085008996712189, 5.34828858761920811791338571452, 6.05521026819151820508235066682, 6.53837769053218520559871320722, 7.15848843667040848804135127934, 7.51147645932417237789624389517, 8.142079716471310775673007806819, 8.721962844816806889443435667476, 9.202662430520918518043337004717, 9.541081905187289685520141516194, 10.38581551283570033324494198450, 10.87760989637962549432871104837, 10.89659567734179704926395564663, 11.49840841845543536633877282910
