| L(s) = 1 | + (−0.597 − 1.28i)2-s + (1.25 − 1.19i)3-s + (−1.28 + 1.53i)4-s + (−0.632 + 1.86i)5-s + (−2.27 − 0.899i)6-s + (−1.03 + 1.35i)7-s + (2.73 + 0.732i)8-s + (0.162 − 2.99i)9-s + (2.76 − 0.302i)10-s + (−5.45 − 0.357i)11-s + (0.208 + 3.45i)12-s + (−4.17 + 4.76i)13-s + (2.35 + 0.522i)14-s + (1.42 + 3.09i)15-s + (−0.694 − 3.93i)16-s + (−2.66 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.422 − 0.906i)2-s + (0.725 − 0.687i)3-s + (−0.642 + 0.766i)4-s + (−0.282 + 0.832i)5-s + (−0.930 − 0.367i)6-s + (−0.392 + 0.511i)7-s + (0.965 + 0.258i)8-s + (0.0541 − 0.998i)9-s + (0.874 − 0.0957i)10-s + (−1.64 − 0.107i)11-s + (0.0601 + 0.998i)12-s + (−1.15 + 1.32i)13-s + (0.629 + 0.139i)14-s + (0.367 + 0.798i)15-s + (−0.173 − 0.984i)16-s + (−0.646 − 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0220 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0220 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.225482 + 0.230509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225482 + 0.230509i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.597 + 1.28i)T \) |
| 3 | \( 1 + (-1.25 + 1.19i)T \) |
| good | 5 | \( 1 + (0.632 - 1.86i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (1.03 - 1.35i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (5.45 + 0.357i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (4.17 - 4.76i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.66 + 2.66i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.0844 - 0.424i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (1.98 - 1.52i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (3.02 - 6.14i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (4.60 + 2.65i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0591 - 0.297i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.21 + 4.76i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (0.447 - 6.82i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (-2.77 + 10.3i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.91 + 5.85i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-3.86 + 11.3i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (-7.79 - 3.84i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.315 - 4.81i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (-3.28 - 7.93i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.23 - 7.80i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (5.51 + 1.47i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-6.39 + 2.17i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (13.2 - 5.50i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (6.93 - 4.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.04810765974148811488347698149, −9.948402603805335335156948083885, −9.291453632597560225989006178210, −8.427911768433170299865847907646, −7.36130085915483746965862837277, −7.00491094694608838576887182407, −5.28249977163590102821557728979, −3.83787184568706955099519430078, −2.68686054549948010413380747270, −2.21789281493443060076473103642,
0.17632113458736526861261546309, 2.53410716119034966965719578427, 4.12063254217972523466767468365, 4.94383403837310268497775176542, 5.71913344142794327993743871060, 7.37633475006050877979598679270, 7.88175659639862810727750540357, 8.581890782862967203697965813152, 9.540865785339542695903225408420, 10.31222242308352830253950632918
