| L(s) = 1 | + 15·4-s − 81·9-s + 224·11-s − 799·16-s − 2.84e3·19-s + 8.30e3·29-s − 1.13e4·31-s − 1.21e3·36-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 + 6.56e3·81-s + 6.77e4·89-s − 1.81e4·99-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 + ⋯ |
| L(s) = 1 | + 0.468·4-s − 1/3·9-s + 0.558·11-s − 0.780·16-s − 1.80·19-s + 1.83·29-s − 2.12·31-s − 0.156·36-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 + 1/9·81-s + 0.906·89-s − 0.186·99-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.085162386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085162386\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 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
−14.43350411535872721683517619118, −13.14803381755329995151211735646, −12.84123689825411315995857025209, −11.98742406625817617132353554742, −11.80640679039483869042524658167, −10.84768055715336950686340022792, −10.76824680821327257780463705348, −10.01542962659354872412841034912, −8.976563486114821189939478229603, −8.935428746105814864313803073967, −8.116028329715209814070841091675, −7.27004530536505098570690947726, −6.71778807415881387422356393942, −6.17544457407817106859297731543, −5.39035791730241333526925251055, −4.41296480902145491353341579345, −3.81251444716435884140696089486, −2.60570089969046927991988659443, −1.96669212965550678468147476781, −0.61791938558354349942937902987,
0.61791938558354349942937902987, 1.96669212965550678468147476781, 2.60570089969046927991988659443, 3.81251444716435884140696089486, 4.41296480902145491353341579345, 5.39035791730241333526925251055, 6.17544457407817106859297731543, 6.71778807415881387422356393942, 7.27004530536505098570690947726, 8.116028329715209814070841091675, 8.935428746105814864313803073967, 8.976563486114821189939478229603, 10.01542962659354872412841034912, 10.76824680821327257780463705348, 10.84768055715336950686340022792, 11.80640679039483869042524658167, 11.98742406625817617132353554742, 12.84123689825411315995857025209, 13.14803381755329995151211735646, 14.43350411535872721683517619118
