| L(s) = 1 | + (0.274 + 1.38i)2-s + (1.54 − 0.779i)3-s + (−1.84 + 0.762i)4-s + (1.10 − 3.26i)5-s + (1.50 + 1.93i)6-s + (0.985 − 1.28i)7-s + (−1.56 − 2.35i)8-s + (1.78 − 2.41i)9-s + (4.83 + 0.640i)10-s + (−4.06 − 0.266i)11-s + (−2.26 + 2.61i)12-s + (−1.99 + 2.27i)13-s + (2.05 + 1.01i)14-s + (−0.828 − 5.91i)15-s + (2.83 − 2.81i)16-s + (1.07 + 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.194 + 0.980i)2-s + (0.893 − 0.449i)3-s + (−0.924 + 0.381i)4-s + (0.495 − 1.45i)5-s + (0.614 + 0.788i)6-s + (0.372 − 0.485i)7-s + (−0.553 − 0.832i)8-s + (0.595 − 0.803i)9-s + (1.52 + 0.202i)10-s + (−1.22 − 0.0804i)11-s + (−0.654 + 0.756i)12-s + (−0.554 + 0.631i)13-s + (0.548 + 0.271i)14-s + (−0.214 − 1.52i)15-s + (0.709 − 0.704i)16-s + (0.261 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95447 - 0.432546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95447 - 0.432546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.274 - 1.38i)T \) |
| 3 | \( 1 + (-1.54 + 0.779i)T \) |
| good | 5 | \( 1 + (-1.10 + 3.26i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (-0.985 + 1.28i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (4.06 + 0.266i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (1.99 - 2.27i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 1.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.23 + 6.19i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.208 + 0.160i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (0.261 - 0.530i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-6.42 - 3.70i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.25 - 11.3i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-9.72 + 7.45i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (0.406 - 6.20i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (-0.799 + 2.98i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.17 - 7.73i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (2.62 - 7.73i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (-4.29 - 2.12i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.0773 - 1.17i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (4.32 + 10.4i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.785 - 1.89i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (5.88 + 1.57i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.58 + 1.21i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (-16.7 + 6.95i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-6.95 + 4.01i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.26988594165027470087363132609, −9.329120437543691893968710373207, −8.784540392907688756409699871009, −7.945040407463259982250066446963, −7.32390388104347361059526774480, −6.15733981760022823823260009699, −4.90807324831983879689606604313, −4.47812900195744771886370960210, −2.78226617775064843669673239137, −1.03396081767006583100030443608,
2.22258612712633995115148905764, 2.71629962954232711536766088041, 3.70494377702829143000328543955, 5.06733019547107155481383888056, 5.92093010770635771415737357093, 7.59927632297038539412428355316, 8.129926752534959448016280805361, 9.474482848770692095146746863707, 10.06432167739122290578429520169, 10.57604436884851686635909147260
