| L(s) = 1 | + 2i·3-s + 4i·5-s − 7-s − 9-s − 8·15-s − 2·17-s − 2i·19-s − 2i·21-s − 8·23-s − 11·25-s + 4i·27-s + 2i·29-s + 4·31-s − 4i·35-s + 6i·37-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + 1.78i·5-s − 0.377·7-s − 0.333·9-s − 2.06·15-s − 0.485·17-s − 0.458i·19-s − 0.436i·21-s − 1.66·23-s − 2.20·25-s + 0.769i·27-s + 0.371i·29-s + 0.718·31-s − 0.676i·35-s + 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9480191156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9480191156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 4iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−9.980844921981508277151230724087, −9.281742392848260667629648452403, −8.213679104104611960617765419335, −7.26321022893750833930218918935, −6.57065360826697413771112406465, −5.87933790607376792102261159123, −4.68810391026922006281793283007, −3.82030343187850065613411227069, −3.15717953483594116481108097039, −2.20731883415605415149215265398,
0.34747576621432203284448000197, 1.41732720483868744900237504857, 2.27308123601168598842906164055, 3.91469022776761970973129087702, 4.63983140838068844909286229286, 5.72755045336466633086239255697, 6.28893585500976620616091365776, 7.32843760445620114999905631088, 8.158787963840575883743100947473, 8.480761702747394361904907518456
