| L(s) = 1 | + (−0.358 + 0.933i)2-s + (0.420 − 0.647i)3-s + (−0.743 − 0.669i)4-s + (1.98 + 1.03i)5-s + (0.453 + 0.624i)6-s + (0.552 + 2.58i)7-s + (0.891 − 0.453i)8-s + (0.977 + 2.19i)9-s + (−1.67 + 1.48i)10-s + (−3.05 + 1.28i)11-s + (−0.745 + 0.199i)12-s + (−6.49 + 1.02i)13-s + (−2.61 − 0.411i)14-s + (1.50 − 0.849i)15-s + (0.104 + 0.994i)16-s + (1.03 + 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 + 0.660i)2-s + (0.242 − 0.373i)3-s + (−0.371 − 0.334i)4-s + (0.886 + 0.461i)5-s + (0.185 + 0.254i)6-s + (0.208 + 0.977i)7-s + (0.315 − 0.160i)8-s + (0.325 + 0.732i)9-s + (−0.529 + 0.468i)10-s + (−0.921 + 0.387i)11-s + (−0.215 + 0.0576i)12-s + (−1.80 + 0.285i)13-s + (−0.698 − 0.110i)14-s + (0.387 − 0.219i)15-s + (0.0261 + 0.248i)16-s + (0.251 + 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563758 + 1.17449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563758 + 1.17449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 - 0.933i)T \) |
| 5 | \( 1 + (-1.98 - 1.03i)T \) |
| 7 | \( 1 + (-0.552 - 2.58i)T \) |
| 11 | \( 1 + (3.05 - 1.28i)T \) |
| good | 3 | \( 1 + (-0.420 + 0.647i)T + (-1.22 - 2.74i)T^{2} \) |
| 13 | \( 1 + (6.49 - 1.02i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 2.69i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (2.93 + 3.25i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.0198 - 0.00532i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.52 + 0.494i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.795 - 0.0836i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.42 + 2.22i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-1.90 - 0.618i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.26 + 5.26i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.358 - 6.84i)T + (-46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (3.22 - 2.61i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.05 + 1.16i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-4.25 + 0.447i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-15.1 - 4.05i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-5.15 + 3.74i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.748 - 14.2i)T + (-72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (0.710 + 1.59i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.37 + 8.67i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.35 + 5.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.67 - 16.9i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−10.28360705925735057474065477992, −9.786914846896770637386719682929, −8.840291522738533933155430678431, −7.920997401606304758393942540137, −7.23310203632032896437616888536, −6.36599301580078684708013918254, −5.29444013996453606435790959848, −4.75639541978942137546720162385, −2.57158453979087365876882693597, −2.06746914085931673665287778697,
0.66723355963714661849519365619, 2.20453115485829543243934957943, 3.30317494618348840943661745842, 4.55132467820897167568052126481, 5.19133917607286305331561135200, 6.57648533022336792601550632521, 7.63905409041602688363124647117, 8.437023951319646210360442141010, 9.576247697583696968193190172060, 9.975027804364326986929852626573
