| L(s) = 1 | + (1.39 + 0.253i)2-s + (−1.72 + 0.169i)3-s + (1.87 + 0.706i)4-s + (−0.715 − 0.260i)5-s + (−2.44 − 0.201i)6-s + (3.32 + 0.586i)7-s + (2.42 + 1.45i)8-s + (2.94 − 0.585i)9-s + (−0.929 − 0.543i)10-s + (1.95 + 5.36i)11-s + (−3.34 − 0.899i)12-s + (−3.36 − 4.01i)13-s + (4.47 + 1.66i)14-s + (1.27 + 0.327i)15-s + (3.00 + 2.64i)16-s + (−2.95 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.983 + 0.179i)2-s + (−0.995 + 0.0981i)3-s + (0.935 + 0.353i)4-s + (−0.319 − 0.116i)5-s + (−0.996 − 0.0820i)6-s + (1.25 + 0.221i)7-s + (0.857 + 0.515i)8-s + (0.980 − 0.195i)9-s + (−0.293 − 0.171i)10-s + (0.588 + 1.61i)11-s + (−0.965 − 0.259i)12-s + (−0.934 − 1.11i)13-s + (1.19 + 0.443i)14-s + (0.329 + 0.0844i)15-s + (0.750 + 0.660i)16-s + (−0.717 + 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64936 + 0.481466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64936 + 0.481466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.253i)T \) |
| 3 | \( 1 + (1.72 - 0.169i)T \) |
| good | 5 | \( 1 + (0.715 + 0.260i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.32 - 0.586i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.95 - 5.36i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.36 + 4.01i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.95 - 1.70i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.947 + 5.37i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.27 + 2.74i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.33 - 0.410i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-9.07 + 5.24i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.88 - 2.24i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.95 - 2.16i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.625 - 3.54i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.06T + 53T^{2} \) |
| 59 | \( 1 + (1.63 - 4.48i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.62 - 0.463i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.16 + 1.81i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.03 + 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.42 - 8.84i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.47 + 6.52i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.79 + 4.49i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.6 + 4.97i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.39601048561890909355841030681, −11.64014916941675060571211199608, −10.88390562661824314736983552637, −9.786058989050329183512099024951, −7.959226945387185966745378924794, −7.20745819565099823961199832372, −5.99797796079973553957161322874, −4.76244183128628570826221099109, −4.40135728417454034828760376535, −2.07276188388821558955609492659,
1.62915622453076410044855522441, 3.78661697487604598691065471229, 4.83413075788252845001742363048, 5.78120596342096898943676395506, 6.91303973111910233128417976128, 7.84428769930359158545173137654, 9.555926679440674660469935050481, 10.97655609162097641270425701567, 11.52778012126581933744740424394, 11.77622611814291732687855573612
