| L(s) = 1 | − i·3-s + i·7-s − 9-s − 4·11-s − 2i·13-s − 2i·17-s + 4·19-s + 21-s + i·27-s + 10·29-s + 4i·33-s − 6i·37-s − 2·39-s − 6·41-s − 4i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s − 0.554i·13-s − 0.485i·17-s + 0.917·19-s + 0.218·21-s + 0.192i·27-s + 1.85·29-s + 0.696i·33-s − 0.986i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8297815901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8297815901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.934549514660148073035207717774, −7.52014669385948410308618551787, −6.68552741638615840665030529596, −5.80000228168391721832248114875, −5.26383555024728082949339658230, −4.45451269102076536026175052377, −3.02631142049500790593921527395, −2.73920277504955962296386464521, −1.47078435540428594792114919328, −0.24135548249415434020466521256,
1.27127149008957950654286886967, 2.58728416385537373803258724699, 3.29203438297566597302190092751, 4.27963955700977926189438003349, 4.93738674114803026501702692710, 5.61568030644377643100687478575, 6.57040054913897829600979146721, 7.23556044699650181363597299832, 8.200643087731278099040476949921, 8.521443766495321936964338769319
