| L(s) = 1 | + (−0.978 − 0.207i)2-s + (1.14 − 1.29i)3-s + (0.913 + 0.406i)4-s + (−2.11 + 0.739i)5-s + (−1.39 + 1.03i)6-s + (−2.09 + 3.62i)7-s + (−0.809 − 0.587i)8-s + (−0.367 − 2.97i)9-s + (2.21 − 0.284i)10-s + (2.20 + 0.468i)11-s + (1.57 − 0.718i)12-s + (5.45 − 1.15i)13-s + (2.80 − 3.11i)14-s + (−1.46 + 3.58i)15-s + (0.669 + 0.743i)16-s + (2.94 + 2.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.662 − 0.749i)3-s + (0.456 + 0.203i)4-s + (−0.943 + 0.330i)5-s + (−0.568 + 0.420i)6-s + (−0.791 + 1.37i)7-s + (−0.286 − 0.207i)8-s + (−0.122 − 0.992i)9-s + (0.701 − 0.0899i)10-s + (0.664 + 0.141i)11-s + (0.454 − 0.207i)12-s + (1.51 − 0.321i)13-s + (0.749 − 0.832i)14-s + (−0.377 + 0.926i)15-s + (0.167 + 0.185i)16-s + (0.713 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07829 + 0.0873394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07829 + 0.0873394i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (-1.14 + 1.29i)T \) |
| 5 | \( 1 + (2.11 - 0.739i)T \) |
| good | 7 | \( 1 + (2.09 - 3.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 0.468i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.45 + 1.15i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.94 - 2.13i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-6.87 - 4.99i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.275 - 0.306i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.639 + 6.08i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.0164 + 0.156i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (3.36 - 10.3i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.114 - 0.0243i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.45 - 4.25i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.521 - 4.96i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (9.89 - 7.18i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.08 + 0.231i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.59 - 1.40i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.632 + 6.01i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 9.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.05 + 6.33i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.0602 - 0.573i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (2.57 - 1.14i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.86 - 5.74i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.56 + 14.8i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.33834095898352369530068002112, −9.937526416481633860640404908640, −9.202227711356803885503892970674, −8.245557188926114158941441677449, −7.84545296803865930585304876464, −6.52211771909459538166904674832, −5.93708874985952746631054814773, −3.58380373129454520408988769731, −3.05967577061492033473590440487, −1.38303381531921321627240009760,
0.948238802905680051643481230845, 3.39902340726834933967698486071, 3.79087748135626890612309709038, 5.20683455294146022782391250950, 6.88747355041805343282819527441, 7.43575179235390597804896506605, 8.545357312875296776671430052934, 9.199003432752778988264937688585, 10.00338991940444498144445702079, 10.99346050954678198193977815123
