| L(s) = 1 | + (−1.68 − 0.898i)3-s + (−1.56 + 1.91i)5-s + (−3.63 − 2.43i)7-s + (0.353 + 0.529i)9-s + (4.39 + 1.33i)11-s + (1.74 − 1.42i)13-s + (4.35 − 1.80i)15-s + (3.75 + 1.55i)17-s + (−0.804 + 8.16i)19-s + (3.93 + 7.36i)21-s + (1.74 + 0.347i)23-s + (−0.217 − 1.09i)25-s + (0.441 + 4.48i)27-s + (0.598 + 1.97i)29-s + (3.81 − 3.81i)31-s + ⋯ |
| L(s) = 1 | + (−0.970 − 0.518i)3-s + (−0.701 + 0.854i)5-s + (−1.37 − 0.919i)7-s + (0.117 + 0.176i)9-s + (1.32 + 0.401i)11-s + (0.483 − 0.396i)13-s + (1.12 − 0.466i)15-s + (0.909 + 0.376i)17-s + (−0.184 + 1.87i)19-s + (0.858 + 1.60i)21-s + (0.364 + 0.0724i)23-s + (−0.0435 − 0.219i)25-s + (0.0850 + 0.863i)27-s + (0.111 + 0.366i)29-s + (0.685 − 0.685i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.684549 + 0.201042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.684549 + 0.201042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (1.68 + 0.898i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (1.56 - 1.91i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (3.63 + 2.43i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.39 - 1.33i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 1.42i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-3.75 - 1.55i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.804 - 8.16i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-1.74 - 0.347i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.598 - 1.97i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-3.81 + 3.81i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.04 - 0.102i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.711 + 3.57i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.793 + 0.423i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.43 - 3.46i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.79 + 5.91i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-4.26 - 3.49i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (2.89 - 5.41i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.61 + 4.89i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (0.916 - 1.37i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.52 + 4.36i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.05 - 9.80i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.630 - 0.0621i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-3.26 + 0.649i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (12.6 - 12.6i)T - 97iT^{2} \) |
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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
−10.95532892068163756858770949027, −10.29921445191837873568892476501, −9.458447138341027235951692867215, −7.981835599521462834018177550296, −7.04126924663490432384620156251, −6.49998095514358663737825090063, −5.78288029323312779481067321593, −3.86704020679096681011239846697, −3.43949877408571173461378848345, −1.07370969730662865065299498785,
0.62316588215023938811021263016, 3.02405596839892287322875326357, 4.22934874553344973229122403914, 5.14698205092377941232737732577, 6.15408340962519964843617957955, 6.82564299358751361402471601685, 8.456605036816878708961885270183, 9.098487976199701925101348913956, 9.801964285382529327753302163504, 11.04831075343197827555854759232
