L(s) = 1 | + i·2-s + i·3-s − 4-s + (1.40 − 1.74i)5-s − 6-s − i·8-s − 9-s + (1.74 + 1.40i)10-s + 1.67·11-s − i·12-s + 4.48i·13-s + (1.74 + 1.40i)15-s + 16-s − 7.61i·17-s − i·18-s + 4.48·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.627 − 0.778i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.550 + 0.443i)10-s + 0.505·11-s − 0.288i·12-s + 1.24i·13-s + (0.449 + 0.362i)15-s + 0.250·16-s − 1.84i·17-s − 0.235i·18-s + 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.863904000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863904000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.40 + 1.74i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.48iT - 13T^{2} \) |
| 17 | \( 1 + 7.61iT - 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 0.482iT - 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 - 3.48T + 31T^{2} \) |
| 37 | \( 1 + 0.482iT - 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 9.57iT - 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.35iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4.51T + 79T^{2} \) |
| 83 | \( 1 - 1.87iT - 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 - 17.7iT - 97T^{2} \) |
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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
−9.369031856481643888005508266903, −9.134814103096724052938441472349, −8.060927085670340726748104766062, −7.12674402835873759033735402680, −6.32045250165647799624219134508, −5.37590887083313327672320317452, −4.76584072817582689844934517498, −3.95994226688819393580873393130, −2.56240057973805297866905676386, −0.990336323856238504529014145970,
1.07991343466882245388602156908, 2.18176813408522913471207217272, 3.12125916206980249582683792413, 3.97883123421067713442361256920, 5.48283828967980429456917003308, 5.98141244023662227906757248540, 6.96418055541242475640467519513, 7.85636271804100105540403561558, 8.607499852350410162753531810577, 9.591862912525457049249391236699