| L(s) = 1 | + (0.123 − 1.40i)2-s + (1.03 − 2.81i)3-s + (−1.96 − 0.347i)4-s + (3.02 + 3.97i)5-s + (−3.84 − 1.80i)6-s + (−1.93 − 1.35i)7-s + (−0.732 + 2.73i)8-s + (−6.86 − 5.82i)9-s + (5.97 − 3.77i)10-s + (−15.0 − 5.47i)11-s + (−3.01 + 5.18i)12-s + (−2.05 − 23.5i)13-s + (−2.14 + 2.56i)14-s + (14.3 − 4.41i)15-s + (3.75 + 1.36i)16-s + (−4.64 − 17.3i)17-s + ⋯ |
| L(s) = 1 | + (0.0616 − 0.704i)2-s + (0.344 − 0.938i)3-s + (−0.492 − 0.0868i)4-s + (0.605 + 0.795i)5-s + (−0.640 − 0.300i)6-s + (−0.276 − 0.193i)7-s + (−0.0915 + 0.341i)8-s + (−0.762 − 0.646i)9-s + (0.597 − 0.377i)10-s + (−1.36 − 0.497i)11-s + (−0.251 + 0.432i)12-s + (−0.158 − 1.81i)13-s + (−0.153 + 0.182i)14-s + (0.955 − 0.294i)15-s + (0.234 + 0.0855i)16-s + (−0.273 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.160641 - 1.36584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160641 - 1.36584i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.123 + 1.40i)T \) |
| 3 | \( 1 + (-1.03 + 2.81i)T \) |
| 5 | \( 1 + (-3.02 - 3.97i)T \) |
| good | 7 | \( 1 + (1.93 + 1.35i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (15.0 + 5.47i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (2.05 + 23.5i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (4.64 + 17.3i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (7.52 - 4.34i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-30.4 + 21.3i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-13.3 - 15.8i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-3.91 + 22.1i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-6.00 - 22.4i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-54.5 - 45.8i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-6.76 - 14.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-16.8 - 11.8i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-61.1 - 61.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (28.7 + 79.0i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-3.63 - 20.6i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (63.4 - 5.55i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-57.9 + 100. i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (1.68 - 6.29i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-14.2 - 16.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-74.9 - 6.56i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (71.6 - 41.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (37.6 - 17.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.05606997422114418366178832419, −10.55237656156467445198880422316, −9.509795683131118300250699399506, −8.271381915619625797867139973442, −7.43438089299715238669682356888, −6.22673076555077289003565271205, −5.19717374648605339016053113119, −2.95446759842965716625680087850, −2.70240039571981072725535893730, −0.63519943790036431641456457416,
2.30345714369969988285798367040, 4.14033607127463246703566757501, 4.97048544504647038141942518244, 5.91143378846175976358995261047, 7.26999879085834554772086302685, 8.578725459580306296363922783094, 9.125010328174957886443411146310, 9.955707039959643264667313514690, 10.95202366659797408655883604761, 12.36764158764278479144167876592
