| L(s) = 1 | + (1.14 − 2.25i)3-s + (−4.68 + 1.73i)5-s + (6.58 − 6.58i)7-s + (1.52 + 2.09i)9-s + (3.81 + 2.77i)11-s + (17.5 + 2.77i)13-s + (−1.47 + 12.5i)15-s + (−7.44 − 14.6i)17-s + (−16.0 − 5.22i)19-s + (−7.28 − 22.4i)21-s + (−5.91 − 37.3i)23-s + (18.9 − 16.2i)25-s + (28.9 − 4.59i)27-s + (−1.46 + 0.477i)29-s + (9.29 − 28.5i)31-s + ⋯ |
| L(s) = 1 | + (0.383 − 0.751i)3-s + (−0.937 + 0.347i)5-s + (0.940 − 0.940i)7-s + (0.169 + 0.232i)9-s + (0.346 + 0.251i)11-s + (1.34 + 0.213i)13-s + (−0.0981 + 0.838i)15-s + (−0.437 − 0.859i)17-s + (−0.846 − 0.275i)19-s + (−0.346 − 1.06i)21-s + (−0.256 − 1.62i)23-s + (0.758 − 0.651i)25-s + (1.07 − 0.170i)27-s + (−0.0506 + 0.0164i)29-s + (0.299 − 0.922i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48451 - 1.12994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48451 - 1.12994i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.68 - 1.73i)T \) |
| good | 3 | \( 1 + (-1.14 + 2.25i)T + (-5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-6.58 + 6.58i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.81 - 2.77i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-17.5 - 2.77i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (7.44 + 14.6i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (16.0 + 5.22i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (5.91 + 37.3i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (1.46 - 0.477i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-9.29 + 28.5i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-1.26 + 8.01i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (30.0 - 21.8i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-25.9 - 25.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (41.1 + 20.9i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-37.0 + 72.8i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (24.7 + 34.1i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-96.0 - 69.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-39.7 - 77.9i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-5.14 - 15.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-9.91 - 62.5i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-19.2 + 6.23i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (22.1 - 11.2i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (39.7 - 54.6i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (26.2 + 13.3i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−11.09562567641855149346194939958, −10.18045326656057721447138859502, −8.517403493739730367648461679710, −8.178064483937303747928627184215, −7.11161286440802526900674975064, −6.58112870605570759165103374439, −4.64601292027001971719235230606, −3.96826257687765464936130291777, −2.33073271128678338674927104058, −0.890418832589694023519107359296,
1.52416171025207891997129439654, 3.45123952975431532755760178156, 4.13341981924258656773215422995, 5.27174930108018250335978576428, 6.42559342597881208755079859373, 7.921367676411719734690944719207, 8.637231978487746068080645217130, 9.074150561757213032554270974966, 10.47411808861039297752125960452, 11.23825920834129109947709960249
