L(s) = 1 | − 1.41·5-s + 2.82i·7-s − 4i·11-s + 2i·13-s + 1.41i·17-s − 5.65·19-s − 4·23-s − 2.99·25-s + 7.07·29-s + 8.48i·31-s − 4.00i·35-s − 8i·37-s − 4.24i·41-s − 11.3·43-s − 12·47-s + ⋯ |
L(s) = 1 | − 0.632·5-s + 1.06i·7-s − 1.20i·11-s + 0.554i·13-s + 0.342i·17-s − 1.29·19-s − 0.834·23-s − 0.599·25-s + 1.31·29-s + 1.52i·31-s − 0.676i·35-s − 1.31i·37-s − 0.662i·41-s − 1.72·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2814590282\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2814590282\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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
−10.27898976266590939030930206800, −9.138578252658849166039969238974, −8.479816739439189293080205960384, −8.041147171731532502990963716136, −6.66945314300088502950600266534, −6.09036336712893615787529073628, −5.08527805937565944259596604056, −4.01888857842834514294148949867, −3.07420847649591753648099150545, −1.85357927114029764483781420041,
0.11657521055228975298798278440, 1.79382330855856724313872427603, 3.21268010215697667350033035799, 4.30907516834417058185510129630, 4.74022134228872666216830827641, 6.26569329355293482022834528538, 6.92635116847851327557431738941, 7.952524465320084492434492596143, 8.200164030092090947921990747998, 9.817759950885973133726472492846