| L(s) = 1 | + (−0.767 − 1.18i)2-s + (0.642 + 0.766i)3-s + (−0.823 + 1.82i)4-s + (−0.425 + 1.16i)5-s + (0.417 − 1.35i)6-s + (−0.404 − 0.233i)7-s + (2.79 − 0.419i)8-s + (−0.173 + 0.984i)9-s + (1.71 − 0.391i)10-s + (0.995 + 1.72i)11-s + (−1.92 + 0.540i)12-s + (−3.03 − 2.54i)13-s + (0.0328 + 0.660i)14-s + (−1.16 + 0.425i)15-s + (−2.64 − 3.00i)16-s + (1.03 + 5.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.542 − 0.840i)2-s + (0.371 + 0.442i)3-s + (−0.411 + 0.911i)4-s + (−0.190 + 0.522i)5-s + (0.170 − 0.551i)6-s + (−0.152 − 0.0883i)7-s + (0.988 − 0.148i)8-s + (−0.0578 + 0.328i)9-s + (0.542 − 0.123i)10-s + (0.300 + 0.520i)11-s + (−0.555 + 0.156i)12-s + (−0.841 − 0.706i)13-s + (0.00876 + 0.176i)14-s + (−0.301 + 0.109i)15-s + (−0.661 − 0.750i)16-s + (0.251 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761870 + 0.481112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761870 + 0.481112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.767 + 1.18i)T \) |
| 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 19 | \( 1 + (2.51 - 3.55i)T \) |
| good | 5 | \( 1 + (0.425 - 1.16i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.404 + 0.233i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.995 - 1.72i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.03 + 2.54i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 5.88i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 3.77i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.00 - 5.67i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.534 + 0.925i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + (2.73 + 3.25i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.67 - 2.43i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-8.83 - 1.55i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.12 + 1.50i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.42 - 0.251i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.734 - 2.01i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.57 - 0.453i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.880 + 0.320i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.54 + 7.17i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 8.95i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.81 + 4.86i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.50 - 4.17i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (11.9 - 2.11i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.77128186231696115078796963630, −10.49795578610474219719313074870, −9.600739081409715434446376351135, −8.699260418373385401879202909617, −7.78632798119311342181249464760, −6.94836245027431532302449267931, −5.30370341168959088794782235333, −3.96266894122569573356209063971, −3.22025945040841774581228161575, −1.85081196426715750096123650858,
0.64860507085233831725474873850, 2.46400911185839850552965854045, 4.34445823970466646759684470408, 5.28425803584301927227342759186, 6.59822054961697364061909937360, 7.16718780835038707168695428280, 8.247497249058220951746541220697, 8.991654826854833536099623354577, 9.570745210802486739121906871543, 10.77275815733019334627690029202
