L(s) = 1 | + 2.56i·3-s − 5.12i·7-s − 3.56·9-s − 4·11-s − 0.561i·13-s + 3.12i·17-s − 4·19-s + 13.1·21-s + i·23-s − 1.43i·27-s + 8.56·29-s + 1.43·31-s − 10.2i·33-s + 7.12i·37-s + 1.43·39-s + ⋯ |
L(s) = 1 | + 1.47i·3-s − 1.93i·7-s − 1.18·9-s − 1.20·11-s − 0.155i·13-s + 0.757i·17-s − 0.917·19-s + 2.86·21-s + 0.208i·23-s − 0.276i·27-s + 1.58·29-s + 0.258·31-s − 1.78i·33-s + 1.17i·37-s + 0.230·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.189732466\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189732466\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 2.56iT - 3T^{2} \) |
| 7 | \( 1 + 5.12iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.561iT - 13T^{2} \) |
| 17 | \( 1 - 3.12iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 7.12iT - 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 9.12iT - 43T^{2} \) |
| 47 | \( 1 - 3.68iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 6.24iT - 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 16.5iT - 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−8.315429839397590504650329507313, −8.155148841363971539368570617805, −7.05996016968879004686350362639, −6.44185048243806787754948264090, −5.26975898131392061225082857582, −4.75650403919347359224137648261, −3.99905550864693684333654420467, −3.57741243969026835730123204641, −2.49784827683864749348359778673, −0.928492083624753121380671307462,
0.38589458470359794684406099841, 1.81256664893155474710845073819, 2.52392607575170497083776337443, 2.89187067048410169559233977724, 4.58019422884141578344468285036, 5.37246112457987608088043507193, 6.00005267268244567192551839797, 6.59780161375034138546394315210, 7.35961324255671109235785818333, 8.276108440104847537546956703852