| L(s) = 1 | + (0.100 + 1.41i)2-s + (2.24 + 0.488i)3-s + (−1.97 + 0.284i)4-s + (−4.50 − 2.17i)5-s + (−0.462 + 3.21i)6-s + (−5.91 + 10.8i)7-s + (−0.601 − 2.76i)8-s + (−3.38 − 1.54i)9-s + (2.61 − 6.57i)10-s + (−2.49 − 2.87i)11-s + (−4.58 − 0.327i)12-s + (−4.98 − 9.12i)13-s + (−15.8 − 7.25i)14-s + (−9.04 − 7.08i)15-s + (3.83 − 1.12i)16-s + (13.1 + 17.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (0.748 + 0.162i)3-s + (−0.494 + 0.0711i)4-s + (−0.900 − 0.434i)5-s + (−0.0770 + 0.536i)6-s + (−0.845 + 1.54i)7-s + (−0.0751 − 0.345i)8-s + (−0.376 − 0.171i)9-s + (0.261 − 0.657i)10-s + (−0.226 − 0.261i)11-s + (−0.381 − 0.0273i)12-s + (−0.383 − 0.702i)13-s + (−1.13 − 0.517i)14-s + (−0.603 − 0.472i)15-s + (0.239 − 0.0704i)16-s + (0.771 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0840792 - 0.567828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0840792 - 0.567828i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.100 - 1.41i)T \) |
| 5 | \( 1 + (4.50 + 2.17i)T \) |
| 23 | \( 1 + (18.0 - 14.2i)T \) |
| good | 3 | \( 1 + (-2.24 - 0.488i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (5.91 - 10.8i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (2.49 + 2.87i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (4.98 + 9.12i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-13.1 - 17.5i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (26.0 - 3.74i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-39.9 - 5.73i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (15.1 + 9.72i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-18.0 - 48.3i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (24.4 + 53.5i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-26.9 - 5.87i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (24.0 - 24.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-46.2 - 25.2i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (25.0 - 85.4i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-63.3 - 40.6i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (9.49 + 132. i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (62.7 - 72.4i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-3.94 + 5.27i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-0.815 + 2.77i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-54.3 + 20.2i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-72.6 - 112. i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (78.8 + 29.4i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−12.44606255251019532236953595071, −11.99585316124068007030563355404, −10.31800013337688738552592643040, −9.170069021634222350076234784211, −8.450099650408613925151711243776, −7.936413581684496480961426631946, −6.27632217437629432048609681649, −5.45552830033051910214574509356, −3.86848967866069150055501651195, −2.79272690721108190319791379906,
0.26504259591241478080155974229, 2.55136520508341107003030188272, 3.62531788873618748183598330577, 4.53622585241572253708095881444, 6.65665000334026225116056411803, 7.53099260756757077908876035325, 8.461746553205346437425887743408, 9.731693563405062175051854827548, 10.48736333809283572770582027810, 11.39334392404167487999777223871
