L(s) = 1 | + (−1.50 − 0.866i)2-s + (−2.59 + 1.5i)3-s + (−2.49 − 4.32i)4-s + (10.6 + 3.36i)5-s + 5.19·6-s + (−14.9 − 10.9i)7-s + 22.5i·8-s + (4.5 − 7.79i)9-s + (−13.0 − 14.2i)10-s + (−16.1 − 28.0i)11-s + (12.9 + 7.49i)12-s + 71.5i·13-s + (12.8 + 29.3i)14-s + (−32.7 + 7.24i)15-s + (−0.474 + 0.821i)16-s + (−79.8 + 46.1i)17-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.306i)2-s + (−0.499 + 0.288i)3-s + (−0.312 − 0.540i)4-s + (0.953 + 0.301i)5-s + 0.353·6-s + (−0.805 − 0.592i)7-s + 0.995i·8-s + (0.166 − 0.288i)9-s + (−0.413 − 0.451i)10-s + (−0.443 − 0.767i)11-s + (0.312 + 0.180i)12-s + 1.52i·13-s + (0.245 + 0.561i)14-s + (−0.563 + 0.124i)15-s + (−0.00740 + 0.0128i)16-s + (−1.13 + 0.657i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0923588 + 0.196916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0923588 + 0.196916i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.59 - 1.5i)T \) |
| 5 | \( 1 + (-10.6 - 3.36i)T \) |
| 7 | \( 1 + (14.9 + 10.9i)T \) |
good | 2 | \( 1 + (1.50 + 0.866i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (16.1 + 28.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 71.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (79.8 - 46.1i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (69.3 - 120. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (104. + 60.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 4.48T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-9.30 - 16.1i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (237. + 136. i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 176. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-151. - 87.6i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (213. - 123. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (381. + 661. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-69.6 + 120. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-133. + 77.2i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 610.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (952. - 550. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (642. - 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 264. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-366. + 635. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 772. iT - 9.12e5T^{2} \) |
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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
−13.83940735916147642905174721628, −12.62386665416871438835888957089, −11.03155535291891645801654827314, −10.44830730224374997466247178965, −9.613393353367441189523094046649, −8.623479581203925193077770324780, −6.56405528625084791030440285629, −5.84814272250238133609868720598, −4.17199052728018971068914993465, −1.91345921906773891040919245782,
0.14524535750147078839043346444, 2.62899627021028212099943578124, 4.84222474653047241458749051159, 6.16416570361804132573193223729, 7.25687280479856735776885796554, 8.631400782008147294352992431369, 9.541924314207695018483070335971, 10.48367613397176964990350350438, 12.15180319892601792430645868931, 13.07195156665425969904151615237