L(s) = 1 | + (−0.786 − 0.618i)2-s + (0.569 − 0.800i)3-s + (0.235 + 0.971i)4-s + (−2.62 + 0.506i)5-s + (−0.942 + 0.276i)6-s + (2.01 + 1.71i)7-s + (0.415 − 0.909i)8-s + (0.665 + 1.92i)9-s + (2.37 + 1.22i)10-s + (2.60 − 2.05i)11-s + (0.912 + 0.365i)12-s + (3.03 − 1.95i)13-s + (−0.524 − 2.59i)14-s + (−1.09 + 2.38i)15-s + (−0.888 + 0.458i)16-s + (3.25 + 3.10i)17-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.437i)2-s + (0.329 − 0.462i)3-s + (0.117 + 0.485i)4-s + (−1.17 + 0.226i)5-s + (−0.384 + 0.112i)6-s + (0.761 + 0.647i)7-s + (0.146 − 0.321i)8-s + (0.221 + 0.640i)9-s + (0.751 + 0.387i)10-s + (0.786 − 0.618i)11-s + (0.263 + 0.105i)12-s + (0.842 − 0.541i)13-s + (−0.140 − 0.693i)14-s + (−0.281 + 0.616i)15-s + (−0.222 + 0.114i)16-s + (0.789 + 0.752i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06738 - 0.218202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06738 - 0.218202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (-2.01 - 1.71i)T \) |
| 23 | \( 1 + (4.75 + 0.631i)T \) |
good | 3 | \( 1 + (-0.569 + 0.800i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (2.62 - 0.506i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 2.05i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.03 + 1.95i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.25 - 3.10i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.92 + 4.69i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (7.60 - 2.23i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.52 + 0.336i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.0260 - 0.0753i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-6.16 - 7.12i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.30 - 7.22i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.14 + 5.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.685 + 14.3i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-1.54 - 0.797i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (2.62 + 3.69i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (9.14 - 3.66i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.604 - 4.20i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.636 + 2.62i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.369 - 7.76i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (10.7 - 12.3i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (9.20 + 0.879i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (9.05 + 10.4i)T + (-13.8 + 96.0i)T^{2} \) |
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
−11.34821318508151859379739756413, −11.04019473399675267737694441621, −9.628778996732649692939935381851, −8.355897301714023487469897391780, −8.108086385468868060533853230757, −7.16301894079054675685375458965, −5.67345668509211300599114742864, −4.10542465218492622452254278015, −2.98346157142160004899694949739, −1.36572995418236743259118167859,
1.22432251948464957904488649461, 3.77534391571789535858169025155, 4.28050085268999195384691498569, 5.86992178028066260014043715815, 7.35658409241638021347564828832, 7.71732416894329343913319182676, 8.898497205288843411262751040722, 9.615247667073339283150353180854, 10.62477880078319613437368103064, 11.82027299353857456108445932998