L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.552 + 3.13i)3-s + (0.766 − 0.642i)4-s + (−0.552 − 3.13i)6-s + (−1.67 − 2.89i)7-s + (−0.500 + 0.866i)8-s + (−6.68 − 2.43i)9-s + (−3.09 + 5.35i)11-s + (1.58 + 2.75i)12-s + (0.128 + 0.727i)13-s + (2.56 + 2.15i)14-s + (0.173 − 0.984i)16-s + (0.815 − 0.296i)17-s + 7.11·18-s + (−2.59 − 3.49i)19-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.318 + 1.80i)3-s + (0.383 − 0.321i)4-s + (−0.225 − 1.27i)6-s + (−0.632 − 1.09i)7-s + (−0.176 + 0.306i)8-s + (−2.22 − 0.810i)9-s + (−0.932 + 1.61i)11-s + (0.458 + 0.794i)12-s + (0.0355 + 0.201i)13-s + (0.685 + 0.574i)14-s + (0.0434 − 0.246i)16-s + (0.197 − 0.0720i)17-s + 1.67·18-s + (−0.596 − 0.802i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.228771 - 0.0859040i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.228771 - 0.0859040i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.59 + 3.49i)T \) |
good | 3 | \( 1 + (0.552 - 3.13i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (1.67 + 2.89i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.09 - 5.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.128 - 0.727i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.815 + 0.296i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.875 + 0.735i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.32 - 2.66i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.23 + 5.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + (-1.68 + 9.54i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.42 + 1.19i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.311 - 0.113i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.44 + 4.56i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (4.56 - 1.66i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 4.64i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.74 - 1.00i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.27 + 4.42i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.35 - 7.69i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.399 - 2.26i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.72 + 9.91i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.404 - 2.29i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.43 - 2.34i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.03433679370647457206183218512, −9.404506751364805007553180768126, −8.560189746448606387078895076540, −7.37042060659115710866063175218, −6.65700337799226005527261557307, −5.39625170542725420528183272123, −4.61950549647394237544448487959, −3.85467473751730982116960791560, −2.56274211763499587632779247290, −0.15925055332147814217579215713,
1.18107622608205720230590842740, 2.52500933729249447796413816065, 3.11194169146032887688460421340, 5.46955996781839657663391019533, 6.04846002070118244947441330813, 6.69742138106273820796021995993, 7.85635702817228863730872322750, 8.326233590897712410076408354156, 8.941984482033014091011647170278, 10.27583137364991838635132322025