| L(s) = 1 | + (0.415 + 1.61i)2-s + (1.28 − 0.244i)3-s + (−0.694 + 0.381i)4-s + (0.929 + 1.97i)6-s + (−0.910 + 1.25i)7-s + (1.38 + 1.47i)8-s + (−1.20 + 0.475i)9-s + (4.66 − 1.19i)11-s + (−0.797 + 0.659i)12-s + (0.940 + 2.37i)13-s + (−2.40 − 0.952i)14-s + (−2.65 + 4.18i)16-s + (0.100 − 0.181i)17-s + (−1.26 − 1.74i)18-s + (0.540 − 2.83i)19-s + ⋯ |
| L(s) = 1 | + (0.293 + 1.14i)2-s + (0.741 − 0.141i)3-s + (−0.347 + 0.190i)4-s + (0.379 + 0.806i)6-s + (−0.344 + 0.473i)7-s + (0.488 + 0.520i)8-s + (−0.400 + 0.158i)9-s + (1.40 − 0.360i)11-s + (−0.230 + 0.190i)12-s + (0.260 + 0.658i)13-s + (−0.642 − 0.254i)14-s + (−0.663 + 1.04i)16-s + (0.0242 − 0.0441i)17-s + (−0.299 − 0.411i)18-s + (0.124 − 0.650i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49997 + 1.80082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49997 + 1.80082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.415 - 1.61i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (-1.28 + 0.244i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (0.910 - 1.25i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-4.66 + 1.19i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.940 - 2.37i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-0.100 + 0.181i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.540 + 2.83i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-3.38 - 0.213i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (4.38 + 0.553i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (7.06 + 3.88i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-9.09 - 5.77i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.203 - 3.24i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 0.423i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-4.98 + 5.30i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (9.10 + 4.28i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 2.09i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.932 + 14.8i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (0.647 + 5.12i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (5.63 + 5.29i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-2.91 - 2.41i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (2.00 + 10.5i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (2.44 + 0.465i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-5.29 + 6.40i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-0.189 + 1.49i)T + (-93.9 - 24.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18550397743293728879609472384, −9.392828373510817544709748310621, −9.036901819642217949357171649239, −8.118805701558788818846309961378, −7.22323591781387081438025867300, −6.38380983237862810302457885178, −5.68743666813787687397720359979, −4.46151882377355349268025365479, −3.25891483105700976881074612853, −1.90083515913746162139515973541,
1.24461366785763000959699111850, 2.60096718081376166960294004865, 3.63073852033256305410624312375, 4.07312529999926086355810506814, 5.69563763738671715458943312253, 6.91180510543565481771517404931, 7.75811480018597719214730890907, 9.058143002366971917402383470238, 9.479264689985693950814230790228, 10.51558821782431422753112181913
