| L(s) = 1 | − 2.82·5-s + i·7-s − 1.41i·11-s − 2i·13-s + 2.82i·17-s + 4·19-s − 1.41·23-s + 3.00·25-s − 1.41·29-s − 2.82i·35-s − 10i·37-s + 5.65i·41-s + 2·43-s − 2.82·47-s − 49-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 0.377i·7-s − 0.426i·11-s − 0.554i·13-s + 0.685i·17-s + 0.917·19-s − 0.294·23-s + 0.600·25-s − 0.262·29-s − 0.478i·35-s − 1.64i·37-s + 0.883i·41-s + 0.304·43-s − 0.412·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7805791592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7805791592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.494644654926225318555982017752, −7.86631687161082124854013279829, −7.44721602386891232782214918690, −6.42608066687600282048957334737, −5.66397755358475809196932631365, −4.89914018381804770602666375844, −3.86984562470340761029369274205, −3.42759838443524989188051426887, −2.36264499740899863886478637161, −0.961249061160140955981512298851,
0.27999804239279711271409905805, 1.56106498477646771898271987144, 2.88214537970609417047787286205, 3.66261365640882094465174384725, 4.41942171824294151749899064245, 5.02435063950794746196635316037, 6.10611313835216158328739420363, 7.08748075004961364540245666117, 7.40611010643688732608778109557, 8.130568469747967142820311220423
