L(s) = 1 | + (1 − i)3-s + (−2 − 2i)5-s + 4i·7-s + i·9-s + (1 + i)11-s + (2 − 2i)13-s − 4·15-s + 4·17-s + (5 − 5i)19-s + (4 + 4i)21-s − 4i·23-s + 3i·25-s + (4 + 4i)27-s + (−6 + 6i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.894 − 0.894i)5-s + 1.51i·7-s + 0.333i·9-s + (0.301 + 0.301i)11-s + (0.554 − 0.554i)13-s − 1.03·15-s + 0.970·17-s + (1.14 − 1.14i)19-s + (0.872 + 0.872i)21-s − 0.834i·23-s + 0.600i·25-s + (0.769 + 0.769i)27-s + (−1.11 + 1.11i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.802667244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802667244\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + 2i)T + 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (-5 + 5i)T - 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (6 - 6i)T - 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-2 - 2i)T + 37iT^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (2 + 2i)T + 53iT^{2} \) |
| 59 | \( 1 + (3 + 3i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6 + 6i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-7 + 7i)T - 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.526817285620139489378377456741, −8.900992957953042825183609476697, −8.187166983210122349020402465142, −7.74548334530815184597716841813, −6.60779301427640553246739078925, −5.38960505960331554720034584180, −4.83029353859977792929546078694, −3.38847437700588197707927242378, −2.47558021980315297150068030380, −1.09494183249180906219837722251,
1.07629744124903481907801786640, 3.13135572245064270892446742574, 3.77906452622817252751683716534, 4.15098899647516312542627175088, 5.80860091744354574857771233461, 6.82688046566606086392714406137, 7.61796878861334286093134699213, 8.095042735430679895525811124551, 9.409920478091233746259209882163, 9.954281070457685068289726245157