L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (1.76 − 2.03i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (1.98 − 1.27i)11-s + (0.841 − 0.540i)12-s + (−1.36 − 1.57i)13-s + (2.57 − 0.757i)14-s + (−0.142 + 0.989i)15-s + (−0.654 + 0.755i)16-s + (1.96 − 4.30i)17-s + ⋯ |
L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (0.169 − 0.371i)6-s + (0.665 − 0.767i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (0.597 − 0.383i)11-s + (0.242 − 0.156i)12-s + (−0.377 − 0.435i)13-s + (0.689 − 0.202i)14-s + (−0.0367 + 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.477 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85789 - 0.740162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85789 - 0.740162i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.956 + 4.69i)T \) |
good | 7 | \( 1 + (-1.76 + 2.03i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.98 + 1.27i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.36 + 1.57i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 4.30i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.47 - 3.22i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.69 + 5.89i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.280 - 1.95i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.79 + 2.28i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.914 + 0.268i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.512 - 3.56i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + (2.57 - 2.96i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.927 + 1.07i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.359 - 2.50i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.29 + 2.76i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.461 - 0.296i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.91 - 8.56i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.555 - 0.640i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (6.18 - 1.81i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.65 - 11.4i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.77 - 0.813i)T + (81.6 + 52.4i)T^{2} \) |
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
−10.56976825762457164343917609041, −9.437946413057009622905133268572, −8.139710377901481256905222730758, −7.76893809367344533318539096221, −6.86694713645471880210157404544, −5.91998567185297687456561911872, −4.85422630345252956542339760703, −3.98464656614349886168231553659, −2.73036620996674859292893676953, −0.975369413382219377401812385896,
1.70950236440145570755051958570, 3.07007526111445141684424021472, 4.13367969898718085741563099152, 4.94703690782182897997700348398, 5.83814042297034599087286703404, 6.93113575061999475747321946236, 8.041608944155772069196276010716, 9.026780119011472703385413401090, 9.778693491434929679131545855262, 10.75920061230020423507116667756