| L(s) = 1 | + (−0.728 + 0.684i)2-s + (0.930 + 1.46i)3-s + (0.0627 − 0.998i)4-s + (0.692 + 2.12i)5-s + (−1.68 − 0.432i)6-s + (−0.369 + 1.13i)7-s + (0.637 + 0.770i)8-s + (−0.00781 + 0.0166i)9-s + (−1.96 − 1.07i)10-s + (1.12 − 1.05i)11-s + (1.52 − 0.836i)12-s + (−2.33 + 4.96i)13-s + (−0.509 − 1.08i)14-s + (−2.47 + 2.99i)15-s + (−0.992 − 0.125i)16-s + (−0.259 − 4.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.515 + 0.484i)2-s + (0.537 + 0.846i)3-s + (0.0313 − 0.499i)4-s + (0.309 + 0.950i)5-s + (−0.686 − 0.176i)6-s + (−0.139 + 0.430i)7-s + (0.225 + 0.272i)8-s + (−0.00260 + 0.00553i)9-s + (−0.619 − 0.340i)10-s + (0.337 − 0.317i)11-s + (0.439 − 0.241i)12-s + (−0.647 + 1.37i)13-s + (−0.136 − 0.289i)14-s + (−0.638 + 0.773i)15-s + (−0.248 − 0.0313i)16-s + (−0.0630 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.609319 + 0.989826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.609319 + 0.989826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.728 - 0.684i)T \) |
| 5 | \( 1 + (-0.692 - 2.12i)T \) |
| good | 3 | \( 1 + (-0.930 - 1.46i)T + (-1.27 + 2.71i)T^{2} \) |
| 7 | \( 1 + (0.369 - 1.13i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 1.05i)T + (0.690 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.33 - 4.96i)T + (-8.28 - 10.0i)T^{2} \) |
| 17 | \( 1 + (0.259 + 4.13i)T + (-16.8 + 2.13i)T^{2} \) |
| 19 | \( 1 + (-1.13 + 1.79i)T + (-8.08 - 17.1i)T^{2} \) |
| 23 | \( 1 + (0.391 - 2.05i)T + (-21.3 - 8.46i)T^{2} \) |
| 29 | \( 1 + (1.21 + 0.479i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (-0.207 - 3.29i)T + (-30.7 + 3.88i)T^{2} \) |
| 37 | \( 1 + (4.62 + 0.584i)T + (35.8 + 9.20i)T^{2} \) |
| 41 | \( 1 + (1.42 + 7.47i)T + (-38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-6.13 - 4.45i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-6.01 + 7.26i)T + (-8.80 - 46.1i)T^{2} \) |
| 53 | \( 1 + (-10.0 + 2.56i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (-0.299 + 0.164i)T + (31.6 - 49.8i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 9.17i)T + (-56.7 - 22.4i)T^{2} \) |
| 67 | \( 1 + (-12.5 + 4.96i)T + (48.8 - 45.8i)T^{2} \) |
| 71 | \( 1 + (-4.07 + 4.93i)T + (-13.3 - 69.7i)T^{2} \) |
| 73 | \( 1 + (9.73 + 5.35i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (-3.27 - 5.15i)T + (-33.6 + 71.4i)T^{2} \) |
| 83 | \( 1 + (2.61 - 4.12i)T + (-35.3 - 75.1i)T^{2} \) |
| 89 | \( 1 + (8.29 + 4.56i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (8.17 + 3.23i)T + (70.7 + 66.4i)T^{2} \) |
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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
−12.10454975415866562873426238661, −11.20046010703319345815878196128, −10.13152004454421383308785446075, −9.356065404088930431241696129425, −8.891283036946281726762392783393, −7.28706451310095184021413954047, −6.60996671217786150625212351626, −5.24573712930697815423112806715, −3.80853386216532326954799803089, −2.42872724231180390847149714491,
1.14294818816535015202590204830, 2.46884295710141872799727783536, 4.12848912755510640322625038076, 5.62490249986014293951293529804, 7.14011670028722304978233546211, 7.993955984312978414331786270185, 8.687664036818719114385132569083, 9.879588084098698713931418062550, 10.55052826882804176232905420637, 12.05726770641932220063725550531
