L(s) = 1 | − 4i·7-s − 4·11-s + 2i·13-s + 6i·17-s + 4·19-s + 2·29-s + 4·31-s − 2i·37-s − 2·41-s − 4i·43-s − 8i·47-s − 9·49-s + 10i·53-s − 4·59-s + 6·61-s + ⋯ |
L(s) = 1 | − 1.51i·7-s − 1.20·11-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 0.371·29-s + 0.718·31-s − 0.328i·37-s − 0.312·41-s − 0.609i·43-s − 1.16i·47-s − 1.28·49-s + 1.37i·53-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.710968545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710968545\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−7.942436900402086903034962248441, −7.13903553524199081565810877067, −6.67226656295563449905007386284, −5.72515634181555500663105018751, −5.01970219584623735485383762292, −4.16671601641457576222334528541, −3.67001684898835786902007793321, −2.68245769799263935496630030517, −1.62615832698289726340825412889, −0.63653679722137863716449576704,
0.68396449882496674321471492532, 2.10052385655208064023860102234, 2.82889786957105707272879037880, 3.24814948191104928071163887097, 4.79590770474619549983835209966, 5.10119823134211183999731131145, 5.75987944408407829034944734107, 6.49678300143192144470358518034, 7.41012102177614558582520485635, 8.030945274866290624391395932123