L(s) = 1 | + (0.00959 − 0.0114i)2-s + (−1.04 − 1.38i)3-s + (0.347 + 1.96i)4-s + (1.26 − 1.05i)5-s + (−0.0258 − 0.00129i)6-s + (1.81 + 1.92i)7-s + (0.0517 + 0.0298i)8-s + (−0.814 + 2.88i)9-s − 0.0245i·10-s + (2.95 − 3.51i)11-s + (2.35 − 2.53i)12-s + (1.52 − 4.19i)13-s + (0.0394 − 0.00230i)14-s + (−2.77 − 0.635i)15-s + (−3.75 + 1.36i)16-s + 3.77·17-s + ⋯ |
L(s) = 1 | + (0.00678 − 0.00808i)2-s + (−0.603 − 0.797i)3-s + (0.173 + 0.984i)4-s + (0.563 − 0.473i)5-s + (−0.0105 − 0.000528i)6-s + (0.686 + 0.727i)7-s + (0.0182 + 0.0105i)8-s + (−0.271 + 0.962i)9-s − 0.00776i·10-s + (0.890 − 1.06i)11-s + (0.680 − 0.732i)12-s + (0.423 − 1.16i)13-s + (0.0105 − 0.000616i)14-s + (−0.717 − 0.163i)15-s + (−0.939 + 0.341i)16-s + 0.915·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19367 - 0.118523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19367 - 0.118523i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.04 + 1.38i)T \) |
| 7 | \( 1 + (-1.81 - 1.92i)T \) |
good | 2 | \( 1 + (-0.00959 + 0.0114i)T + (-0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 1.05i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-2.95 + 3.51i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.52 + 4.19i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 1.46iT - 19T^{2} \) |
| 23 | \( 1 + (2.54 - 6.98i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (2.83 + 7.78i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (8.27 - 1.45i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.397 + 0.688i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.97 + 1.44i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.303 - 1.71i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.286 - 1.62i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.66 + 2.11i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.8 + 3.95i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.96 - 0.522i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.30 - 6.96i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.66 + 3.84i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 7.91i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.22 - 1.02i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.61 - 1.68i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 + (-1.56 - 0.276i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.47346835353423082201774036653, −11.68192749320483318357210680258, −11.03853723228283597070312234596, −9.337778293316655379934010434783, −8.232516608585780814236877561465, −7.63864573639360881733791160591, −6.02185659635284394707350406406, −5.42833516579746877713901206672, −3.44339928074997859590381428925, −1.66200606636842504355266634588,
1.66914650186055983857923072327, 4.08011699255654412420452870170, 5.04082481497808894311286387731, 6.31370019616711021508512460661, 7.01875013086572439078374518408, 9.027353313161516459586703988045, 9.858697120668681063073382457485, 10.60283263435900059498547675037, 11.29562315442919268404939466922, 12.33029171524768363878329649840