L(s) = 1 | − 2.82i·7-s − 5.65·11-s − 2i·13-s + 2i·17-s + 2.82i·23-s + 6·29-s − 5.65·31-s + 10i·37-s − 2·41-s + 8.48i·43-s + 2.82i·47-s − 1.00·49-s − 6i·53-s − 11.3·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 1.06i·7-s − 1.70·11-s − 0.554i·13-s + 0.485i·17-s + 0.589i·23-s + 1.11·29-s − 1.01·31-s + 1.64i·37-s − 0.312·41-s + 1.29i·43-s + 0.412i·47-s − 0.142·49-s − 0.824i·53-s − 1.47·59-s − 0.256·61-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
\(L(1)\) |
\(\approx\) |
\(1.266166508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.266166508\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.921925426099951868678946160967, −7.45506270835760872358901200625, −6.61005727794237701068526685585, −5.88511205878476361218286964804, −5.01141070697186690976656219957, −4.57668138792436507615124268942, −3.45821674675477819568040917461, −2.95501577763324480433586879451, −1.82703938522790392996229776317, −0.72303286314502468889046315025,
0.41996540591292181751682563827, 2.04335727876463045059910511894, 2.50122322779165823029852736069, 3.35003215205336487942276309473, 4.42741845741385986054532101570, 5.19218545621568710898378051732, 5.61633048900528609860934201220, 6.42764372962017001071778309508, 7.29380197993992485935342729542, 7.82094732656024622994843459021