L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.15 + 1.29i)3-s + 1.00i·4-s + (0.616 + 0.616i)5-s + (0.0958 − 1.72i)6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.331 + 2.98i)9-s − 0.872i·10-s + (−2.63 + 2.63i)11-s + (−1.29 + 1.15i)12-s + (1.08 + 3.43i)13-s − 1.00i·14-s + (−0.0836 + 1.50i)15-s − 1.00·16-s − 5.36·17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.666 + 0.745i)3-s + 0.500i·4-s + (0.275 + 0.275i)5-s + (0.0391 − 0.706i)6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.110 + 0.993i)9-s − 0.275i·10-s + (−0.793 + 0.793i)11-s + (−0.372 + 0.333i)12-s + (0.300 + 0.953i)13-s − 0.267i·14-s + (−0.0215 + 0.389i)15-s − 0.250·16-s − 1.30·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05010 + 0.822951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05010 + 0.822951i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.15 - 1.29i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (-1.08 - 3.43i)T \) |
good | 5 | \( 1 + (-0.616 - 0.616i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.63 - 2.63i)T - 11iT^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 + (-3.60 + 3.60i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 + 3.00iT - 29T^{2} \) |
| 31 | \( 1 + (6.60 - 6.60i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.44 - 4.44i)T + 37iT^{2} \) |
| 41 | \( 1 + (-8.43 - 8.43i)T + 41iT^{2} \) |
| 43 | \( 1 - 0.610iT - 43T^{2} \) |
| 47 | \( 1 + (-3.46 + 3.46i)T - 47iT^{2} \) |
| 53 | \( 1 + 0.464iT - 53T^{2} \) |
| 59 | \( 1 + (-8.93 + 8.93i)T - 59iT^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 + (-1.77 + 1.77i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.00980 + 0.00980i)T + 71iT^{2} \) |
| 73 | \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + (3.29 + 3.29i)T + 83iT^{2} \) |
| 89 | \( 1 + (-8.90 + 8.90i)T - 89iT^{2} \) |
| 97 | \( 1 + (5.41 - 5.41i)T - 97iT^{2} \) |
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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
−10.96345061772333167894515203246, −9.970755930724341880864972485019, −9.313332316465626317442451128241, −8.631881042874069946131760673005, −7.64137541515448533998881798451, −6.66217587943788132003156455518, −5.04264416884544036285826906477, −4.27136432929459096935850333185, −2.85172969219578143731868924427, −2.04720580251779214286725564249,
0.853258555836415953098221298151, 2.33245740099804142203600963811, 3.69295183430930119394483898496, 5.37594809148990858531109895280, 6.06675392388215935811635238330, 7.40327788925210884138580009330, 7.78191044339802057075987228783, 8.820409914094011563469917102553, 9.331015074712709659737009129114, 10.62105732337916407365370320027