L(s) = 1 | + (−1.65 − 1.65i)5-s − 7-s + (0.993 − 0.993i)11-s + (2.62 + 2.62i)13-s − 2.77i·17-s + (−1.56 + 1.56i)19-s + 1.05i·23-s + 0.491i·25-s + (3.47 − 3.47i)29-s + 1.06i·31-s + (1.65 + 1.65i)35-s + (−0.0657 + 0.0657i)37-s + 6.31·41-s + (2.38 + 2.38i)43-s + 1.47·47-s + ⋯ |
L(s) = 1 | + (−0.741 − 0.741i)5-s − 0.377·7-s + (0.299 − 0.299i)11-s + (0.728 + 0.728i)13-s − 0.673i·17-s + (−0.358 + 0.358i)19-s + 0.219i·23-s + 0.0982i·25-s + (0.645 − 0.645i)29-s + 0.190i·31-s + (0.280 + 0.280i)35-s + (−0.0108 + 0.0108i)37-s + 0.985·41-s + (0.364 + 0.364i)43-s + 0.214·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.097697352\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097697352\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (1.65 + 1.65i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.993 + 0.993i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.62 - 2.62i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.77iT - 17T^{2} \) |
| 19 | \( 1 + (1.56 - 1.56i)T - 19iT^{2} \) |
| 23 | \( 1 - 1.05iT - 23T^{2} \) |
| 29 | \( 1 + (-3.47 + 3.47i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.06iT - 31T^{2} \) |
| 37 | \( 1 + (0.0657 - 0.0657i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 + (-2.38 - 2.38i)T + 43iT^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + (7.63 + 7.63i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.15 + 4.15i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.78 + 7.78i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.98 - 1.98i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 9.50iT - 73T^{2} \) |
| 79 | \( 1 - 9.85iT - 79T^{2} \) |
| 83 | \( 1 + (1.13 + 1.13i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.04T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−8.251525131939462300682962216450, −7.61855224847689593719114793847, −6.64669305287690372370869646405, −6.12231307248142144448581387870, −5.11265075424814277895089051361, −4.29660689256743173541367549963, −3.76151758725359021621303743991, −2.73118433618980860955347271339, −1.45806762928041246259225830986, −0.36835501851288102039301171742,
1.10160626949482344769297383339, 2.48334420805933749804018656527, 3.32329317624028099415861503859, 3.94863763989858136991493618883, 4.81760478539520796601869093086, 5.97296369873004281842333645455, 6.40931165747377994361529072165, 7.33038063638632771137248481569, 7.78236322160435729889830105464, 8.703677991825013504911564430843