L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 5.46i·5-s + 2.65·7-s − 2.82i·8-s + 7.72·10-s − 3.75i·11-s − 23.2·13-s + 3.75i·14-s + 4.00·16-s + 19.2i·17-s − 35.0·19-s + 10.9i·20-s + 5.30·22-s − 4.79i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.09i·5-s + 0.379·7-s − 0.353i·8-s + 0.772·10-s − 0.341i·11-s − 1.79·13-s + 0.268i·14-s + 0.250·16-s + 1.13i·17-s − 1.84·19-s + 0.546i·20-s + 0.241·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.206413 - 0.398760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.206413 - 0.398760i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 4.79iT \) |
good | 5 | \( 1 + 5.46iT - 25T^{2} \) |
| 7 | \( 1 - 2.65T + 49T^{2} \) |
| 11 | \( 1 + 3.75iT - 121T^{2} \) |
| 13 | \( 1 + 23.2T + 169T^{2} \) |
| 17 | \( 1 - 19.2iT - 289T^{2} \) |
| 19 | \( 1 + 35.0T + 361T^{2} \) |
| 29 | \( 1 + 45.2iT - 841T^{2} \) |
| 31 | \( 1 + 32.4T + 961T^{2} \) |
| 37 | \( 1 + 40.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 57.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 35.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 60.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 139.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 67.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 45.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 12.2T + 9.40e3T^{2} \) |
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
−10.52249365836831893831506036517, −9.613059764971445512775037651257, −8.584681174250991573768176977169, −8.115046614309739516758803308785, −6.95678932199288956841868753488, −5.83151992407176262508354916041, −4.87380038826724202787998048836, −4.10154230236255697900699517846, −2.05833968378287567437363198667, −0.17955061693212614178267244711,
2.08245145594237024206829123234, 2.95269534637573615309017136002, 4.36184910116213132784599436303, 5.34056022103408834016884151628, 6.89932988292400423865900833923, 7.43496106968487091407712670042, 8.810235968094859194846479669990, 9.704939752619201716151085901760, 10.62760600217910343851395325346, 11.07950976603268623037893593672