| L(s) = 1 | + (−1.19 − 1.60i)2-s + (0.412 − 1.53i)3-s + (−1.13 + 3.83i)4-s + (3.53 + 3.54i)5-s + (−2.95 + 1.18i)6-s + (0.696 − 6.96i)7-s + (7.50 − 2.78i)8-s + (5.59 + 3.23i)9-s + (1.44 − 9.89i)10-s + (3.33 − 1.92i)11-s + (5.43 + 3.32i)12-s + (6.09 − 6.09i)13-s + (−11.9 + 7.22i)14-s + (6.90 − 3.97i)15-s + (−13.4 − 8.68i)16-s + (1.46 − 5.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.598 − 0.800i)2-s + (0.137 − 0.512i)3-s + (−0.282 + 0.959i)4-s + (0.706 + 0.708i)5-s + (−0.493 + 0.197i)6-s + (0.0995 − 0.995i)7-s + (0.937 − 0.347i)8-s + (0.621 + 0.359i)9-s + (0.144 − 0.989i)10-s + (0.302 − 0.174i)11-s + (0.453 + 0.276i)12-s + (0.468 − 0.468i)13-s + (−0.856 + 0.516i)14-s + (0.460 − 0.264i)15-s + (−0.839 − 0.542i)16-s + (0.0861 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01269 - 0.794189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01269 - 0.794189i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 + 1.60i)T \) |
| 5 | \( 1 + (-3.53 - 3.54i)T \) |
| 7 | \( 1 + (-0.696 + 6.96i)T \) |
| good | 3 | \( 1 + (-0.412 + 1.53i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-3.33 + 1.92i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-6.09 + 6.09i)T - 169iT^{2} \) |
| 17 | \( 1 + (-1.46 + 5.46i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (8.53 + 4.92i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.92 + 18.3i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 33.2iT - 841T^{2} \) |
| 31 | \( 1 + (-24.1 - 41.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-13.8 - 51.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 9.69iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-9.19 + 9.19i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.02 - 15.0i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (2.53 - 9.47i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (54.0 - 31.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (12.2 + 7.09i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (28.3 - 105. i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 44.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-4.30 - 1.15i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (66.8 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (109. + 109. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-77.1 + 133. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (123. + 123. i)T + 9.40e3iT^{2} \) |
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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
−12.79894858313473311778477282335, −11.48947349966040677681309479204, −10.44540105114287266601456299840, −10.01809914113187256629399803299, −8.525013128657303713993257384826, −7.44445517995606615474981118405, −6.52623003122830698572528323175, −4.37662596743064159481684848022, −2.81229361150715672092039237648, −1.27649029020812864412841957368,
1.67232655318549906362191844452, 4.32901258208669295187326852898, 5.56589341889219929881932158517, 6.50052194456763071635765745539, 8.084896319937304068316853191615, 9.207843692034571218140544249618, 9.492042763844431953715979598294, 10.75143843636248964273188897779, 12.21075545087489962700993055876, 13.27399333660244668161615117998
