| L(s) = 1 | + 2i·7-s + 11-s − i·13-s − i·17-s − 4·19-s − i·23-s + 5·29-s + 31-s + 6i·37-s − 7i·43-s + 7i·47-s + 3·49-s + 12i·53-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s + 0.301·11-s − 0.277i·13-s − 0.242i·17-s − 0.917·19-s − 0.208i·23-s + 0.928·29-s + 0.179·31-s + 0.986i·37-s − 1.06i·43-s + 1.02i·47-s + 0.428·49-s + 1.64i·53-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709975122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709975122\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 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
−8.385505387719458446938286564844, −7.67070689557175807334461262451, −6.72779144001320854807441256832, −6.22505549548480915382681375905, −5.41465136428094452442428538847, −4.68230215194648102845723579985, −3.86943734200286838567029651915, −2.85643756423026950406725372067, −2.19035269333625169688436513937, −0.961687572878066953455925939998,
0.53453983323481518663677549440, 1.66932950422626434459552698595, 2.63368115076741450350968166267, 3.75833817454951547784460863793, 4.20806667609395150143893487289, 5.09962748974859434969231042613, 5.95743199083639168433177707665, 6.78995086703178448939484440777, 7.13515111669526287765585740295, 8.224321525524111180145217636878
