| L(s) = 1 | + (−0.587 + 0.809i)2-s + (1.50 + 2.59i)3-s + (0.927 + 2.85i)4-s + (−3.88 − 5.35i)5-s + (−2.98 − 0.303i)6-s + (2.73 + 8.42i)7-s + (−6.65 − 2.16i)8-s + (−4.44 + 7.82i)9-s + 6.61·10-s + (−6 + 6.70i)12-s + (−10.7 − 7.77i)13-s + (−8.42 − 2.73i)14-s + (8.01 − 18.1i)15-s + (−4.04 + 2.93i)16-s + (−3.52 − 4.85i)17-s + (−3.71 − 8.19i)18-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.404i)2-s + (0.502 + 0.864i)3-s + (0.231 + 0.713i)4-s + (−0.777 − 1.07i)5-s + (−0.497 − 0.0506i)6-s + (0.390 + 1.20i)7-s + (−0.832 − 0.270i)8-s + (−0.494 + 0.869i)9-s + 0.661·10-s + (−0.5 + 0.559i)12-s + (−0.823 − 0.598i)13-s + (−0.601 − 0.195i)14-s + (0.534 − 1.21i)15-s + (−0.252 + 0.183i)16-s + (−0.207 − 0.285i)17-s + (−0.206 − 0.455i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.183597 - 0.788784i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183597 - 0.788784i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.50 - 2.59i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.587 - 0.809i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (3.88 + 5.35i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (-2.73 - 8.42i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (10.7 + 7.77i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (3.52 + 4.85i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (2.81 - 8.67i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 - 17.5iT - 529T^{2} \) |
| 29 | \( 1 + (25.1 - 8.15i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-5.01 - 3.64i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (6.05 + 18.6i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-5.32 - 1.72i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.8 - 5.47i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (13.0 - 18.0i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-20.8 + 6.77i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (75.5 - 54.8i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 76.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + (38.6 + 53.2i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (4.64 + 14.2i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-95.5 - 69.3i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-65.3 - 89.9i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 97.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-98.0 - 71.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−11.97333982197949615052803283862, −10.86353753565547987896564768148, −9.364882469109258387252247023732, −8.951672352614622713960729711260, −8.077552260980376156900461645464, −7.55713616599222122368795207391, −5.71086163212973447210742608446, −4.81413135991137314271478719612, −3.68009565774097883106351302957, −2.45208600824561780185174962645,
0.35566638566095665706264477729, 1.91421865351865709675656275078, 3.08645633535326335199908581920, 4.41175519137631510980357046468, 6.22525558152013356782899854697, 7.08725914241424645489569864365, 7.58518392531875013393886580804, 8.829853501656210535895588459548, 9.943804008479464020091009572560, 10.83704168622827086780611284633
