| L(s) = 1 | − i·3-s − 2i·7-s − 9-s + 11-s − 8i·17-s + 8·19-s − 2·21-s − 4i·23-s + i·27-s + 6·29-s − i·33-s + 6i·37-s − 2·41-s − 2i·43-s − 4i·47-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.755i·7-s − 0.333·9-s + 0.301·11-s − 1.94i·17-s + 1.83·19-s − 0.436·21-s − 0.834i·23-s + 0.192i·27-s + 1.11·29-s − 0.174i·33-s + 0.986i·37-s − 0.312·41-s − 0.304i·43-s − 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042533486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042533486\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 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
−7.61839975375642601302324527177, −7.00190949550921785669093930579, −6.70883040759150333342930480010, −5.56868606029200738402527268971, −4.99235516590005760948603119913, −4.17367255792814338863269024038, −3.14119440342357911374726679775, −2.59232967348681751526873376582, −1.23882157580044949334876803780, −0.59895102003418578682316630807,
1.16042143986838353801238030376, 2.16431386271444548306017826196, 3.21614378908363478835842933461, 3.75097053385369957048358600555, 4.66882319100171949225243533082, 5.48297414910302520296289474791, 5.92862147463046249414199029981, 6.72237497000611550431506246506, 7.70244499336281913361375763585, 8.218521616980910281565085030975
