L(s) = 1 | − 3.04·5-s − 0.418i·7-s − 1.27i·11-s − 3.88·17-s + 4.35i·19-s − 4i·23-s + 4.27·25-s + 1.27i·35-s + 5.67i·43-s − 2.72i·47-s + 6.82·49-s + 3.88i·55-s + 11.2·61-s + 5.82·73-s − 0.533·77-s + ⋯ |
L(s) = 1 | − 1.36·5-s − 0.158i·7-s − 0.384i·11-s − 0.941·17-s + 0.999i·19-s − 0.834i·23-s + 0.854·25-s + 0.215i·35-s + 0.865i·43-s − 0.397i·47-s + 0.974·49-s + 0.523i·55-s + 1.44·61-s + 0.681·73-s − 0.0608·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005664092\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005664092\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 + 0.418iT - 7T^{2} \) |
| 11 | \( 1 + 1.27iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.67iT - 43T^{2} \) |
| 47 | \( 1 + 2.72iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.579176938916276012275428290126, −8.246291911068748907521177931678, −7.42405470766442011919906115737, −6.73150062419532408906159378567, −5.87128302284941346032846861853, −4.78815285518097073447780197133, −4.06322748888702814141512599074, −3.42354428583469254161262216129, −2.26744060251389066631068576766, −0.74352218561140812312628572624,
0.49958246135212697847096695750, 2.07899172260403764149102275518, 3.16363193364839340689103775054, 4.04751486265555312745144796946, 4.67179823649224692980629021085, 5.59087212461064865477261991178, 6.73600505732201445227103114343, 7.25252361581787737498463160277, 7.964452556271228295761384415273, 8.746879286977994964254811811477