L(s) = 1 | − 1.43i·2-s + 3i·3-s + 5.93·4-s + 4.31·6-s − 7i·7-s − 20.0i·8-s − 9·9-s + 7.61·11-s + 17.7i·12-s − 52.3i·13-s − 10.0·14-s + 18.6·16-s − 49.7i·17-s + 12.9i·18-s − 140.·19-s + ⋯ |
L(s) = 1 | − 0.508i·2-s + 0.577i·3-s + 0.741·4-s + 0.293·6-s − 0.377i·7-s − 0.885i·8-s − 0.333·9-s + 0.208·11-s + 0.428i·12-s − 1.11i·13-s − 0.192·14-s + 0.290·16-s − 0.709i·17-s + 0.169i·18-s − 1.69·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.756216714\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.756216714\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 2 | \( 1 + 1.43iT - 8T^{2} \) |
| 11 | \( 1 - 7.61T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 49.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 115. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 322. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 76.6iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 424. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 107.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 915.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 451. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 907.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 755. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 22.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 549. iT - 9.12e5T^{2} \) |
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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
−10.40675316361608626941193161411, −9.557175131080434770539843799139, −8.451108413335784301587286484962, −7.43023294125820936438976674549, −6.48265574308458684825144417685, −5.44616821086103789804405193978, −4.15923711529089998816759776223, −3.22456397247156522018546168291, −2.09159333960394245954018345206, −0.48899603228758433131339986425,
1.63780101468500349214939641569, 2.49264503942889908297833096289, 4.06709527517205054360542815519, 5.43872052990465124462062814153, 6.54189239517683060237137939159, 6.73876828635332411885197724590, 8.099929963366335391685513309538, 8.577770142033226400463424398554, 9.813164182012394865446654893605, 10.99364174527801513598750779898