L(s) = 1 | + 2i·7-s − 4·11-s + 6i·13-s − 2i·17-s + 8·19-s + 6i·23-s − 2·29-s − 4·31-s + 2i·37-s + 10·41-s − 2i·43-s − 2i·47-s + 3·49-s + 2i·53-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.755i·7-s − 1.20·11-s + 1.66i·13-s − 0.485i·17-s + 1.83·19-s + 1.25i·23-s − 0.371·29-s − 0.718·31-s + 0.328i·37-s + 1.56·41-s − 0.304i·43-s − 0.291i·47-s + 0.428·49-s + 0.274i·53-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
\(L(1)\) |
\(\approx\) |
\(1.099663475\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099663475\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.158612445167268624467718886067, −7.35370810152734424801121162544, −7.11870392947495710033762548875, −5.88589679689211788197868551223, −5.49469499764820767726993150029, −4.79859793478613078513193858421, −3.87514303327217382414160328483, −2.97906650141699390657940763340, −2.26481536708461324525101001005, −1.30406220189609918803480510026,
0.28592301097037861580262170974, 1.16197170673049304313712587365, 2.54712089899293853042849582176, 3.12047531341511215748495110627, 3.96901847562351030726427160329, 4.88289730872330907037856433912, 5.54146736187135672687687523372, 6.03985797920729168638768491608, 7.28727733510542347683748946493, 7.52485320010173922110264104675