L(s) = 1 | + (−3.16 − 1.82i)2-s + (−2.61 − 1.47i)3-s + (4.66 + 8.07i)4-s + (1.93 + 1.11i)5-s + (5.58 + 9.42i)6-s + 7·7-s − 19.4i·8-s + (4.67 + 7.69i)9-s + (−4.08 − 7.06i)10-s + (2.13 − 1.23i)11-s + (−0.312 − 27.9i)12-s − 14.8·13-s + (−22.1 − 12.7i)14-s + (−3.41 − 5.77i)15-s + (−16.8 + 29.1i)16-s + (14.2 − 8.23i)17-s + ⋯ |
L(s) = 1 | + (−1.58 − 0.912i)2-s + (−0.871 − 0.490i)3-s + (1.16 + 2.01i)4-s + (0.387 + 0.223i)5-s + (0.930 + 1.57i)6-s + 7-s − 2.42i·8-s + (0.519 + 0.854i)9-s + (−0.408 − 0.706i)10-s + (0.194 − 0.112i)11-s + (−0.0260 − 2.33i)12-s − 1.13·13-s + (−1.58 − 0.912i)14-s + (−0.227 − 0.384i)15-s + (−1.05 + 1.82i)16-s + (0.839 − 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0744 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0744 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.414640 - 0.384823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.414640 - 0.384823i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.61 + 1.47i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 + (3.16 + 1.82i)T + (2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 1.23i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 + (-14.2 + 8.23i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-15.5 + 26.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-21.5 - 12.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 24.9iT - 841T^{2} \) |
| 31 | \( 1 + (-14.0 - 24.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-19.0 + 33.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 20.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (6.66 + 3.85i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (68.4 - 39.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-67.1 + 38.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 18.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-42.3 - 73.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 5.61iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (24.9 + 43.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.2 - 66.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 1.56iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (1.19 + 0.688i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 71.8T + 9.40e3T^{2} \) |
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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
−12.72367523005414888199670751148, −11.67861656771764119711065994217, −11.22790011451318903607630534470, −10.14958208838617059304599784692, −9.200023784450573862631586334533, −7.74188106527757337310219478527, −7.07898295801606534789953228005, −5.12932122817840771206006950109, −2.50665782845480013706956407281, −0.975636051505353616182408518033,
1.27710069863986634218519970507, 4.96954816841510097755296732189, 5.95078559806246175147707833888, 7.25592999050920234251874632667, 8.287751768667670031834567502238, 9.603962869991060719647523272182, 10.16783651998284785933515340356, 11.25301213779803119687481620991, 12.33487358578404561187886195230, 14.51760413618507624173420911126