| L(s) = 1 | + 2i·7-s − 6i·13-s + 7i·17-s − 7·19-s − 7i·23-s + 6·29-s + 3·31-s + 6i·37-s − 4·41-s + 8i·43-s − 4i·47-s + 3·49-s + 5i·53-s + 6·59-s − 3·61-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 1.66i·13-s + 1.69i·17-s − 1.60·19-s − 1.45i·23-s + 1.11·29-s + 0.538·31-s + 0.986i·37-s − 0.624·41-s + 1.21i·43-s − 0.583i·47-s + 0.428·49-s + 0.686i·53-s + 0.781·59-s − 0.384·61-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
| \(L(1)\) |
\(\approx\) |
\(1.032641157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032641157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 7iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.394790549707488788573337380341, −8.043248232173501152491445748750, −6.78598149572076604114647486990, −6.20487615250579627771017281003, −5.67379391896086907151472198942, −4.72115242419042266201577303505, −4.03801953946096459850782512882, −2.93547215739186280656298788038, −2.37021235446739421869703039101, −1.13314663382230015992141297643,
0.28324131138333249347033449932, 1.57932595799940519617239884297, 2.45224142994588789549267618635, 3.53301384129525085486638733689, 4.33059965308362548978086102828, 4.81841450888981401610263871052, 5.84059655059319666092373621237, 6.78679112981941549774513036328, 7.03795585566005738998142211646, 7.84916255871418029656914020428
