| L(s) = 1 | + (−0.100 − 1.41i)2-s + (0.265 + 0.0578i)3-s + (−1.97 + 0.284i)4-s + (3.98 + 3.01i)5-s + (0.0547 − 0.380i)6-s + (−2.60 + 4.77i)7-s + (0.601 + 2.76i)8-s + (−8.11 − 3.70i)9-s + (3.85 − 5.92i)10-s + (12.2 + 14.1i)11-s + (−0.542 − 0.0388i)12-s + (3.94 + 7.23i)13-s + (7.00 + 3.19i)14-s + (0.885 + 1.03i)15-s + (3.83 − 1.12i)16-s + (8.34 + 11.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (0.0886 + 0.0192i)3-s + (−0.494 + 0.0711i)4-s + (0.797 + 0.603i)5-s + (0.00912 − 0.0634i)6-s + (−0.372 + 0.682i)7-s + (0.0751 + 0.345i)8-s + (−0.902 − 0.411i)9-s + (0.385 − 0.592i)10-s + (1.11 + 1.28i)11-s + (−0.0452 − 0.00323i)12-s + (0.303 + 0.556i)13-s + (0.500 + 0.228i)14-s + (0.0590 + 0.0688i)15-s + (0.239 − 0.0704i)16-s + (0.490 + 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46079 + 0.375887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46079 + 0.375887i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (-3.98 - 3.01i)T \) |
| 23 | \( 1 + (-10.9 + 20.2i)T \) |
| good | 3 | \( 1 + (-0.265 - 0.0578i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (2.60 - 4.77i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-12.2 - 14.1i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-3.94 - 7.23i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-8.34 - 11.1i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (27.0 - 3.89i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-50.8 - 7.31i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-24.2 - 15.6i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (18.0 + 48.4i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (8.69 + 19.0i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-25.6 - 5.57i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (19.9 - 19.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (74.7 + 40.8i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (19.5 - 66.6i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (39.7 + 25.5i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (4.39 + 61.4i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (55.5 - 64.0i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-69.3 + 92.6i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (28.8 - 98.2i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-16.2 + 6.07i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (60.3 + 93.8i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (71.4 + 26.6i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−12.27683086223531137321979480957, −11.02218484221703313636523419888, −10.15160103112786659345198797369, −9.213003467000032742438255887921, −8.582029735579940041377681200470, −6.70107874801358414459520252699, −6.05470123839319332656479104435, −4.43170568799395499300829597233, −3.00186444753123702608793675544, −1.85028891223101753212404806665,
0.875896144967083748336863271911, 3.16038666619443524281329851772, 4.71116577378883311994881224481, 5.92448682174192414350105759048, 6.56250669128546121154959404221, 8.167046600057005966211172271274, 8.745238282127981138039214549022, 9.774695255393700113658776109319, 10.81302460564955898837648506883, 11.95868749606623477320938333268
