| L(s) = 1 | + (0.221 + 1.39i)2-s + (−5.16 − 2.63i)3-s + (−1.90 + 0.618i)4-s + (−4.87 + 1.09i)5-s + (2.53 − 7.79i)6-s + (−2.21 − 2.21i)7-s + (−1.28 − 2.52i)8-s + (14.4 + 19.9i)9-s + (−2.61 − 6.57i)10-s + (−4.40 − 3.19i)11-s + (11.4 + 1.81i)12-s + (1.11 − 7.05i)13-s + (2.60 − 3.58i)14-s + (28.0 + 7.15i)15-s + (3.23 − 2.35i)16-s + (−21.2 + 10.8i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (−1.72 − 0.877i)3-s + (−0.475 + 0.154i)4-s + (−0.975 + 0.219i)5-s + (0.422 − 1.29i)6-s + (−0.316 − 0.316i)7-s + (−0.160 − 0.315i)8-s + (1.60 + 2.21i)9-s + (−0.261 − 0.657i)10-s + (−0.400 − 0.290i)11-s + (0.954 + 0.151i)12-s + (0.0858 − 0.542i)13-s + (0.185 − 0.255i)14-s + (1.87 + 0.477i)15-s + (0.202 − 0.146i)16-s + (−1.24 + 0.635i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0151416 - 0.0625936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0151416 - 0.0625936i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.221 - 1.39i)T \) |
| 5 | \( 1 + (4.87 - 1.09i)T \) |
| good | 3 | \( 1 + (5.16 + 2.63i)T + (5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (2.21 + 2.21i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.40 + 3.19i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-1.11 + 7.05i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (21.2 - 10.8i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (14.7 + 4.78i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (0.196 - 0.0311i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (12.3 - 4.02i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-6.10 + 18.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-1.82 - 0.288i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (8.77 - 6.37i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (12.7 - 12.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (36.7 - 72.1i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-30.3 - 15.4i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-21.0 - 28.9i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (47.9 + 34.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (10.9 - 5.58i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (17.0 + 52.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-74.3 + 11.7i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (132. - 43.0i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (11.4 + 22.4i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-77.0 + 106. i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-23.6 + 46.3i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−15.23140224207734672612036823735, −13.30517635433726936829735377285, −12.66719123776498218311434265974, −11.38186805253777599412104167783, −10.56468064188895888641543926869, −8.117564946161524845266611589378, −7.01361833548485253358146101045, −6.07108115252457517192723640728, −4.53564670839499262855417490593, −0.07582887225596499180929465657,
4.04932533625893728866697619202, 5.07795137198748449822674208852, 6.70344528900570872330507714501, 9.018668184403047903253078864410, 10.31907051917213454563130935180, 11.28850154770243260247525816242, 11.97390142725380918362639757792, 12.94373650389444143389310441853, 15.11297676954737782975361220107, 15.88155926917417216341900431919
