L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.864 + 0.864i)5-s + (−2.02 + 1.70i)7-s + (0.707 − 0.707i)8-s + 1.22·10-s + (3.50 − 3.50i)11-s + (3.37 + 1.25i)13-s + (2.63 + 0.222i)14-s − 1.00·16-s + 0.322·17-s + (−1.77 + 1.77i)19-s + (−0.864 − 0.864i)20-s − 4.95·22-s − 2.70i·23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.386 + 0.386i)5-s + (−0.763 + 0.645i)7-s + (0.250 − 0.250i)8-s + 0.386·10-s + (1.05 − 1.05i)11-s + (0.937 + 0.348i)13-s + (0.704 + 0.0593i)14-s − 0.250·16-s + 0.0781·17-s + (−0.406 + 0.406i)19-s + (−0.193 − 0.193i)20-s − 1.05·22-s − 0.564i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004675281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004675281\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.02 - 1.70i)T \) |
| 13 | \( 1 + (-3.37 - 1.25i)T \) |
good | 5 | \( 1 + (0.864 - 0.864i)T - 5iT^{2} \) |
| 11 | \( 1 + (-3.50 + 3.50i)T - 11iT^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 + (1.77 - 1.77i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.70iT - 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + (-1.72 + 1.72i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.27 + 2.27i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.46 - 6.46i)T - 41iT^{2} \) |
| 43 | \( 1 + 0.393iT - 43T^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 + (-10.7 - 10.7i)T + 59iT^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-5.38 - 5.38i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.647 - 0.647i)T + 71iT^{2} \) |
| 73 | \( 1 + (-6.85 - 6.85i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.81T + 79T^{2} \) |
| 83 | \( 1 + (6.09 - 6.09i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.68 - 3.68i)T + 89iT^{2} \) |
| 97 | \( 1 + (5.20 - 5.20i)T - 97iT^{2} \) |
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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
−9.392706628246377133439537381880, −8.766338907887197584892274203388, −8.222382379829303411672550948583, −7.04026185382298730489556270769, −6.35386169738805885357670600291, −5.62435093515332933732304772536, −3.98255338742910584107044327869, −3.53820394966285664404923181467, −2.49259523678054783738941736073, −1.10804022053835977901716522133,
0.55158528342482315144556995103, 1.81787317726074546486800321538, 3.50722537720726918058196148133, 4.20274641173527079596670949216, 5.21990956727873518616655984608, 6.40255578879861606905086071568, 6.80105436393383463957278776988, 7.69448722582461031269889737105, 8.480301904220075072256094268155, 9.271608313466336372958883640975