L(s) = 1 | + (0.902 − 1.08i)2-s + (−0.736 + 0.425i)3-s + (−0.372 − 1.96i)4-s + (−0.166 + 0.166i)5-s + (−0.201 + 1.18i)6-s + (0.684 + 2.55i)7-s + (−2.47 − 1.36i)8-s + (−1.13 + 1.97i)9-s + (0.0311 + 0.331i)10-s + (−1.39 − 0.373i)11-s + (1.10 + 1.28i)12-s + (−0.406 − 3.58i)13-s + (3.39 + 1.55i)14-s + (0.0517 − 0.193i)15-s + (−3.72 + 1.46i)16-s + (1.21 + 0.699i)17-s + ⋯ |
L(s) = 1 | + (0.637 − 0.770i)2-s + (−0.425 + 0.245i)3-s + (−0.186 − 0.982i)4-s + (−0.0744 + 0.0744i)5-s + (−0.0821 + 0.483i)6-s + (0.258 + 0.965i)7-s + (−0.875 − 0.483i)8-s + (−0.379 + 0.657i)9-s + (0.00984 + 0.104i)10-s + (−0.419 − 0.112i)11-s + (0.320 + 0.371i)12-s + (−0.112 − 0.993i)13-s + (0.908 + 0.416i)14-s + (0.0133 − 0.0498i)15-s + (−0.930 + 0.366i)16-s + (0.293 + 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.884760 - 0.359864i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.884760 - 0.359864i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.902 + 1.08i)T \) |
| 13 | \( 1 + (0.406 + 3.58i)T \) |
good | 3 | \( 1 + (0.736 - 0.425i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.166 - 0.166i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.684 - 2.55i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.39 + 0.373i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.21 - 0.699i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.39 + 1.44i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (4.37 + 7.57i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.11 + 3.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.88 - 3.88i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.133 - 0.5i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.59 - 1.5i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.59 - 7.95i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.80 - 2.80i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 + (-2.20 - 8.22i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.61 + 6.25i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.652 - 2.43i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-10.5 + 2.81i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-5.05 - 5.05i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.51iT - 79T^{2} \) |
| 83 | \( 1 + (6.91 + 6.91i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.71 - 6.41i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.00 + 14.9i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−15.22644166680972392498029064681, −14.13608954756364912772996122779, −12.90275643991810611044429470437, −11.84687966499133623718359761856, −10.92229209462279933245393432347, −9.819607770888714046590442615080, −8.185497935699895824222560428914, −5.87446296878319027077752243767, −4.95410484897495864656367425494, −2.77888614299363005415744565685,
3.82459082718375572087543020344, 5.44305012992469908334693829028, 6.84643491907153342298170014350, 7.894459550268062722109946463175, 9.573608136655472773389887446063, 11.43789546245764113671427838300, 12.23945391615291463406412552991, 13.67464146434701479985949323528, 14.30069004587196368325740548077, 15.69139687905799627949458503758