L(s) = 1 | + (0.179 + 1.40i)2-s − i·3-s + (−1.93 + 0.502i)4-s − i·5-s + (1.40 − 0.179i)6-s − 1.34·7-s + (−1.05 − 2.62i)8-s − 9-s + (1.40 − 0.179i)10-s + 0.705·11-s + (0.502 + 1.93i)12-s − 3.25·13-s + (−0.240 − 1.88i)14-s − 15-s + (3.49 − 1.94i)16-s + 4.04i·17-s + ⋯ |
L(s) = 1 | + (0.126 + 0.991i)2-s − 0.577i·3-s + (−0.967 + 0.251i)4-s − 0.447i·5-s + (0.572 − 0.0730i)6-s − 0.507·7-s + (−0.371 − 0.928i)8-s − 0.333·9-s + (0.443 − 0.0566i)10-s + 0.212·11-s + (0.144 + 0.558i)12-s − 0.903·13-s + (−0.0642 − 0.503i)14-s − 0.258·15-s + (0.873 − 0.486i)16-s + 0.981i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8604068140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8604068140\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.179 - 1.40i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-2.51 - 4.08i)T \) |
good | 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 0.705T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 4.04iT - 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 1.38iT - 31T^{2} \) |
| 37 | \( 1 - 8.57iT - 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 3.29iT - 47T^{2} \) |
| 53 | \( 1 - 7.62iT - 53T^{2} \) |
| 59 | \( 1 - 6.68iT - 59T^{2} \) |
| 61 | \( 1 - 9.90iT - 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 + 8.64iT - 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 - 4.44T + 83T^{2} \) |
| 89 | \( 1 + 7.34iT - 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.467142817183666547383964300510, −9.086697630271112547147738436282, −7.988895846826962303832621747512, −7.47994328534477234302413677474, −6.65823965911271230375304147007, −5.82604197091081652327072682290, −5.11875033611435142267243783415, −4.04751376991570957459561771874, −3.00128121221889902176444837186, −1.28945723230490355909570382123,
0.36218615716224458033296608526, 2.22971225611040368877345249088, 3.08208972152539864052405629987, 3.90263160483048484217248397477, 4.92381422006129858483077714227, 5.61834196978382357025583276629, 6.83006862143715242887717225635, 7.73474104192900994498956845914, 8.885493645409994392765797972142, 9.655312338122863104084084586372