| L(s) = 1 | + (−1.89 + 0.625i)2-s + (2.48 − 1.67i)3-s + (3.21 − 2.37i)4-s + (1.04 + 5.95i)5-s + (−3.67 + 4.74i)6-s + (1.18 − 1.41i)7-s + (−4.62 + 6.52i)8-s + (3.36 − 8.34i)9-s + (−5.71 − 10.6i)10-s + (14.9 + 2.63i)11-s + (4.01 − 11.3i)12-s + (−6.81 − 2.48i)13-s + (−1.36 + 3.42i)14-s + (12.5 + 13.0i)15-s + (4.69 − 15.2i)16-s + (2.22 + 3.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 + 0.312i)2-s + (0.828 − 0.559i)3-s + (0.804 − 0.594i)4-s + (0.209 + 1.19i)5-s + (−0.612 + 0.790i)6-s + (0.169 − 0.201i)7-s + (−0.577 + 0.816i)8-s + (0.374 − 0.927i)9-s + (−0.571 − 1.06i)10-s + (1.36 + 0.239i)11-s + (0.334 − 0.942i)12-s + (−0.524 − 0.190i)13-s + (−0.0977 + 0.244i)14-s + (0.839 + 0.869i)15-s + (0.293 − 0.955i)16-s + (0.130 + 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25735 + 0.167328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25735 + 0.167328i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.89 - 0.625i)T \) |
| 3 | \( 1 + (-2.48 + 1.67i)T \) |
| good | 5 | \( 1 + (-1.04 - 5.95i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-1.18 + 1.41i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-14.9 - 2.63i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (6.81 + 2.48i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-2.22 - 3.85i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-30.1 - 17.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.19 + 9.77i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (26.3 - 9.59i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (34.0 + 40.5i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-1.20 - 2.08i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (44.2 + 16.1i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (15.3 + 2.71i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (13.3 - 15.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 77.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (69.2 - 12.2i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-25.1 - 21.0i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-25.3 + 69.6i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-81.7 + 47.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (42.1 - 72.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (0.961 + 2.64i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (19.3 + 53.0i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-55.7 + 96.5i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.0 - 73.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−14.05632806121261437573253021337, −12.29144626365845110089553126397, −11.26341337298496878817089162572, −9.965086650452080993904923806078, −9.274514441342201324163489120128, −7.81580151022520373738491746770, −7.12581381832692083992042118692, −6.10488232232208427863757213248, −3.35522468356031344646502344531, −1.74706871406649695116323645629,
1.56689088137226714302644544962, 3.46907499262097437913711584065, 5.09658814921383625217472262290, 7.14194507384525577956127258232, 8.389594581650513305074199941207, 9.277784274065161309400333523619, 9.635963866427689245882583748619, 11.26847780075768660465320247255, 12.17209366385346387347324012052, 13.37793083759396912049136272639
