| L(s) = 1 | + (0.156 − 0.987i)2-s + (2.92 − 0.463i)3-s + (−0.951 − 0.309i)4-s + (0.685 + 2.12i)5-s − 2.96i·6-s + (0.0732 + 0.143i)7-s + (−0.453 + 0.891i)8-s + (5.48 − 1.78i)9-s + (2.20 − 0.343i)10-s + (−1.56 − 0.509i)11-s + (−2.92 − 0.463i)12-s + (−3.20 + 0.507i)13-s + (0.153 − 0.0498i)14-s + (2.99 + 5.90i)15-s + (0.809 + 0.587i)16-s + (2.49 − 4.89i)17-s + ⋯ |
| L(s) = 1 | + (0.110 − 0.698i)2-s + (1.68 − 0.267i)3-s + (−0.475 − 0.154i)4-s + (0.306 + 0.951i)5-s − 1.20i·6-s + (0.0276 + 0.0543i)7-s + (−0.160 + 0.315i)8-s + (1.82 − 0.594i)9-s + (0.698 − 0.108i)10-s + (−0.472 − 0.153i)11-s + (−0.844 − 0.133i)12-s + (−0.889 + 0.140i)13-s + (0.0409 − 0.0133i)14-s + (0.772 + 1.52i)15-s + (0.202 + 0.146i)16-s + (0.604 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04046 - 0.802531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04046 - 0.802531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 + 0.987i)T \) |
| 5 | \( 1 + (-0.685 - 2.12i)T \) |
| 31 | \( 1 + (3.00 + 4.68i)T \) |
| good | 3 | \( 1 + (-2.92 + 0.463i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.0732 - 0.143i)T + (-4.11 + 5.66i)T^{2} \) |
| 11 | \( 1 + (1.56 + 0.509i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (3.20 - 0.507i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 4.89i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.749 - 1.03i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.04 + 0.534i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (8.50 - 6.17i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (2.89 + 2.89i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.990 + 0.719i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (0.849 - 5.36i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-10.9 + 1.73i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (5.53 + 2.81i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.15 + 1.58i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 - 14.1iT - 61T^{2} \) |
| 67 | \( 1 + (1.04 - 1.04i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.03 + 9.34i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.16 - 6.20i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-3.67 - 11.3i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.27 - 8.02i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.87 + 15.0i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.63 - 9.09i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.59248161745192089152773340165, −10.45117990230599065014688937184, −9.632317053509224749418843750447, −9.013881249627861347422682509594, −7.67675884521867055371199609770, −7.19629679066670606148087245044, −5.43940072244225971380402267961, −3.78946887163707818577603286899, −2.86700336271375277531014021944, −2.06875410695430054910668105860,
2.03959561384104516220868810433, 3.57050335140696539114795568533, 4.62035192164635323102883198650, 5.74161113677904482167708756465, 7.44823681214587565276685324564, 7.980261925759254569031714415298, 8.899106225504019643869184815479, 9.542679946477620572377239228227, 10.38243941637672610391747490786, 12.30325302742163716938301993852
