| 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 | \( 1 - 12.2T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07950976603268623037893593672, −10.62760600217910343851395325346, −9.704939752619201716151085901760, −8.810235968094859194846479669990, −7.43496106968487091407712670042, −6.89932988292400423865900833923, −5.34056022103408834016884151628, −4.36184910116213132784599436303, −2.95269534637573615309017136002, −2.08245145594237024206829123234,
0.17955061693212614178267244711, 2.05833968378287567437363198667, 4.10154230236255697900699517846, 4.87380038826724202787998048836, 5.83151992407176262508354916041, 6.95678932199288956841868753488, 8.115046614309739516758803308785, 8.584681174250991573768176977169, 9.613059764971445512775037651257, 10.52249365836831893831506036517
