| L(s) = 1 | + (−0.105 + 0.0186i)2-s + (−0.492 + 2.79i)3-s + (−1.86 + 0.680i)4-s − 0.304i·6-s + (3.01 + 2.52i)7-s + (0.370 − 0.214i)8-s + (−4.73 − 1.72i)9-s + (2.81 + 4.86i)11-s + (−0.979 − 5.55i)12-s + (0.765 + 2.10i)13-s + (−0.365 − 0.210i)14-s + (3.01 − 2.52i)16-s + (−2.08 + 5.73i)17-s + (0.532 + 0.0939i)18-s + (3.23 + 0.571i)19-s + ⋯ |
| L(s) = 1 | + (−0.0747 + 0.0131i)2-s + (−0.284 + 1.61i)3-s + (−0.934 + 0.340i)4-s − 0.124i·6-s + (1.13 + 0.955i)7-s + (0.131 − 0.0756i)8-s + (−1.57 − 0.574i)9-s + (0.847 + 1.46i)11-s + (−0.282 − 1.60i)12-s + (0.212 + 0.582i)13-s + (−0.0976 − 0.0563i)14-s + (0.752 − 0.631i)16-s + (−0.505 + 1.38i)17-s + (0.125 + 0.0221i)18-s + (0.743 + 0.131i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0281596 - 1.26061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0281596 - 1.26061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + (5.63 + 2.28i)T \) |
| good | 2 | \( 1 + (0.105 - 0.0186i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.492 - 2.79i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-3.01 - 2.52i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.81 - 4.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.765 - 2.10i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.08 - 5.73i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-3.23 - 0.571i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.91 - 1.10i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.38 + 0.798i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.45iT - 31T^{2} \) |
| 41 | \( 1 + (-8.92 + 3.24i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 12.5iT - 43T^{2} \) |
| 47 | \( 1 + (-3.68 + 6.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 1.73i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.823 + 0.981i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.313 + 0.861i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.06 - 1.73i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.38 + 7.85i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 + (5.87 - 6.99i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 0.771i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (0.660 + 0.786i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (8.14 + 4.70i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.33222972484461397812964134588, −9.554235789172287473488928038268, −8.951694673357651689380914509018, −8.462860270068045650091827618250, −7.22245649439303850810121861929, −5.73247029306188145635196805111, −5.04277994339427107807517697376, −4.26618848564884204514680880031, −3.77455074572652636786678231769, −1.95030239169807739345399005110,
0.833671574012128386685867406355, 1.18921936515734802489630976734, 3.03048022682845205135568924104, 4.44593835248671735133630855835, 5.37437124952593307363717238299, 6.25885785036820807003398884654, 7.20310197861413176509799839547, 7.947058802642361168992017032563, 8.577031138776659356917351739537, 9.459011971293659132285063702354
