L(s) = 1 | + (−0.726 − 2.23i)2-s + (−2.85 + 2.07i)4-s + (−2.13 − 0.694i)5-s + (−2.38 − 3.27i)7-s + (2.90 + 2.10i)8-s + 5.28i·10-s + (3.31 + 0.0200i)11-s + (4.42 − 1.43i)13-s + (−5.59 + 7.70i)14-s + (0.427 − 1.31i)16-s + (0.0235 − 0.0725i)17-s + (1.40 − 1.93i)19-s + (7.53 − 2.44i)20-s + (−2.36 − 7.42i)22-s − 3.22i·23-s + ⋯ |
L(s) = 1 | + (−0.513 − 1.58i)2-s + (−1.42 + 1.03i)4-s + (−0.956 − 0.310i)5-s + (−0.899 − 1.23i)7-s + (1.02 + 0.745i)8-s + 1.67i·10-s + (0.999 + 0.00604i)11-s + (1.22 − 0.398i)13-s + (−1.49 + 2.05i)14-s + (0.106 − 0.328i)16-s + (0.00571 − 0.0175i)17-s + (0.323 − 0.444i)19-s + (1.68 − 0.547i)20-s + (−0.504 − 1.58i)22-s − 0.672i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.000880781 - 0.568205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000880781 - 0.568205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.31 - 0.0200i)T \) |
good | 2 | \( 1 + (0.726 + 2.23i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (2.13 + 0.694i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (2.38 + 3.27i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-4.42 + 1.43i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.0235 + 0.0725i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.40 + 1.93i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 3.22iT - 23T^{2} \) |
| 29 | \( 1 + (1.48 - 1.08i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.517 - 1.59i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.87 - 4.27i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.82 - 4.96i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.28iT - 43T^{2} \) |
| 47 | \( 1 + (-3.65 + 5.02i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.16 - 0.379i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.341 - 0.469i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.59 + 1.16i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + (1.06 + 0.346i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.82 - 10.7i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.627 + 0.203i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.15 + 9.71i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 6.58iT - 89T^{2} \) |
| 97 | \( 1 + (-5.08 - 15.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.03546170021631902199781450422, −12.11986899844358967670279363634, −11.18008677430688746017387332576, −10.37661599593391814969964488364, −9.274735293173336602986796233262, −8.232672145853905120298687386574, −6.72540443750132085188246097818, −4.12774478154439322216338921883, −3.43283514169905169840176564669, −0.855652997329293444855679185769,
3.73667479960224348845956228086, 5.73717545520714436180078245035, 6.53124109005508026100759207173, 7.69750669254702180655329149614, 8.838386943022131644960127744666, 9.463378423603972416847587503809, 11.31763942722825184534517235082, 12.33090517676581196959596342690, 13.82385061502975220616048624790, 14.82329736041423648954689826432