L(s) = 1 | + (1 + i)3-s + (−1 + i)5-s − 2i·7-s − i·9-s + (−1 + i)11-s + (−1 − i)13-s − 2·15-s − 2·17-s + (−3 − 3i)19-s + (2 − 2i)21-s + 6i·23-s + 3i·25-s + (4 − 4i)27-s + (3 + 3i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s + (−0.447 + 0.447i)5-s − 0.755i·7-s − 0.333i·9-s + (−0.301 + 0.301i)11-s + (−0.277 − 0.277i)13-s − 0.516·15-s − 0.485·17-s + (−0.688 − 0.688i)19-s + (0.436 − 0.436i)21-s + 1.25i·23-s + 0.600i·25-s + (0.769 − 0.769i)27-s + (0.557 + 0.557i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.943451 + 0.187664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.943451 + 0.187664i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 5 | \( 1 + (1 - i)T - 5iT^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-3 - 3i)T + 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3 + 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (9 + 9i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5 - 5i)T + 67iT^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-1 - i)T + 83iT^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−15.15806687836610164216618137463, −14.08609491620803761202087545846, −12.95059353984951873190801474811, −11.48778254946706635368882741265, −10.38797986358525437670973004875, −9.337445096027797491465193578904, −7.943227555938203405118009237953, −6.71055742683799316053347084224, −4.56124503667803827231168171669, −3.19980240997241295571649335002,
2.46719287874121892924646954170, 4.66177529723384040571861019470, 6.44149975752932406076517563358, 8.094575571028396639247074109329, 8.623372325765915389171432951705, 10.29444745064215992476530998944, 11.79680949406565484562975769216, 12.68493327599984495986802133024, 13.71646814124117271064522680575, 14.80890398541489430227393698201