L(s) = 1 | + (0.221 + 1.39i)2-s + (−1.54 − 0.786i)3-s + (−1.90 + 0.618i)4-s + (3.55 − 3.51i)5-s + (0.756 − 2.32i)6-s + (−5.48 − 5.48i)7-s + (−1.28 − 2.52i)8-s + (1.76 + 2.42i)9-s + (5.69 + 4.19i)10-s + (−11.2 − 8.18i)11-s + (3.42 + 0.541i)12-s + (1.89 − 11.9i)13-s + (6.44 − 8.87i)14-s + (−8.25 + 2.62i)15-s + (3.23 − 2.35i)16-s + (27.6 − 14.1i)17-s + ⋯ |
L(s) = 1 | + (0.110 + 0.698i)2-s + (−0.514 − 0.262i)3-s + (−0.475 + 0.154i)4-s + (0.711 − 0.702i)5-s + (0.126 − 0.388i)6-s + (−0.783 − 0.783i)7-s + (−0.160 − 0.315i)8-s + (0.195 + 0.269i)9-s + (0.569 + 0.419i)10-s + (−1.02 − 0.743i)11-s + (0.285 + 0.0451i)12-s + (0.146 − 0.921i)13-s + (0.460 − 0.633i)14-s + (−0.550 + 0.174i)15-s + (0.202 − 0.146i)16-s + (1.62 − 0.829i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.904500 - 0.543009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.904500 - 0.543009i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.221 - 1.39i)T \) |
| 3 | \( 1 + (1.54 + 0.786i)T \) |
| 5 | \( 1 + (-3.55 + 3.51i)T \) |
good | 7 | \( 1 + (5.48 + 5.48i)T + 49iT^{2} \) |
| 11 | \( 1 + (11.2 + 8.18i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 11.9i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-27.6 + 14.1i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-25.8 - 8.41i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (30.6 - 4.84i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (51.9 - 16.8i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (0.628 - 1.93i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-30.9 - 4.90i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (30.7 - 22.3i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-0.187 + 0.187i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-26.5 + 52.1i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (5.62 + 2.86i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-5.07 - 6.98i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-87.6 - 63.6i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-7.98 + 4.06i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-39.4 - 121. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 2.17i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-40.6 + 13.2i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (18.4 + 36.1i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-79.2 + 109. i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-24.2 + 47.6i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.98197271117037808875292421144, −11.83451371535673624792854847043, −10.23326079126802833540491735824, −9.720344895931871231848851316152, −8.114151757045475419118755633716, −7.30886143547320177494210394959, −5.70655862777857228705300603646, −5.42742855173189090823543984180, −3.42894877817053749863892775667, −0.70588495284707676281570944882,
2.15405431903788370965651550593, 3.55577729577751909982542833011, 5.34810832295419308824108624072, 6.12676516451674675610042860240, 7.63618680888654604513517570737, 9.557847654201523964296508067819, 9.786109239424148701134714626225, 10.90573896458234369134904794525, 11.97534645789228270075858228103, 12.74227452630488770674576699014