L(s) = 1 | − 1.73i·3-s + 3.46i·7-s − 2.99·9-s − 2·13-s − 3.46i·19-s + 5.99·21-s + 5·25-s + 5.19i·27-s − 10.3i·31-s − 10·37-s + 3.46i·39-s + 10.3i·43-s − 4.99·49-s − 5.99·57-s + 14·61-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + 1.30i·7-s − 0.999·9-s − 0.554·13-s − 0.794i·19-s + 1.30·21-s + 25-s + 0.999i·27-s − 1.86i·31-s − 1.64·37-s + 0.554i·39-s + 1.58i·43-s − 0.714·49-s − 0.794·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.771843 - 0.206814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.771843 - 0.206814i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−15.44375138708661754247537486481, −14.46956971764483229019315726323, −13.12925026034704855482017874281, −12.24056695896816489745402282108, −11.26412486568634322020379293105, −9.338892356501978764549936387737, −8.201099013122993350293745119564, −6.74418422965018797428766625394, −5.37448383935061293331068662521, −2.52494733590377164458054694339,
3.63537616080026223751227237238, 5.06935218801281721891351547462, 7.02451456717377461766522669630, 8.625402204712350869792689549676, 10.10407073322686381731412723868, 10.73331120502494611207295480445, 12.23659147913522161754220310176, 13.83864131223898957951710912710, 14.60504122510036768778392573017, 15.92369172058273067985032747780