L(s) = 1 | + (−0.763 − 1.19i)2-s + (−0.835 + 1.81i)4-s + (1.27 − 1.83i)5-s + 1.23i·7-s + (2.80 − 0.391i)8-s + (−3.16 − 0.116i)10-s + 4.64i·11-s − 4.85·13-s + (1.46 − 0.940i)14-s + (−2.60 − 3.03i)16-s − 0.206i·17-s + 7.40i·19-s + (2.27 + 3.85i)20-s + (5.52 − 3.54i)22-s − 1.04i·23-s + ⋯ |
L(s) = 1 | + (−0.539 − 0.841i)2-s + (−0.417 + 0.908i)4-s + (0.570 − 0.821i)5-s + 0.465i·7-s + (0.990 − 0.138i)8-s + (−0.999 − 0.0367i)10-s + 1.39i·11-s − 1.34·13-s + (0.392 − 0.251i)14-s + (−0.651 − 0.759i)16-s − 0.0500i·17-s + 1.69i·19-s + (0.508 + 0.861i)20-s + (1.17 − 0.754i)22-s − 0.218i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7061184894\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7061184894\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.763 + 1.19i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.27 + 1.83i)T \) |
good | 7 | \( 1 - 1.23iT - 7T^{2} \) |
| 11 | \( 1 - 4.64iT - 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 0.206iT - 17T^{2} \) |
| 19 | \( 1 - 7.40iT - 19T^{2} \) |
| 23 | \( 1 + 1.04iT - 23T^{2} \) |
| 29 | \( 1 - 6.52iT - 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + 4.41iT - 47T^{2} \) |
| 53 | \( 1 - 3.80T + 53T^{2} \) |
| 59 | \( 1 - 5.04iT - 59T^{2} \) |
| 61 | \( 1 - 7.09iT - 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 - 8.71iT - 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 7.15T + 89T^{2} \) |
| 97 | \( 1 - 5.82iT - 97T^{2} \) |
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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
−10.16413435395155433330630377176, −9.253979708634928603481115039448, −8.707103686859766538208430701956, −7.68819034355704163689748089578, −6.95470262748490376738633651286, −5.42384912699140030322682035733, −4.82274260538682611866263864446, −3.71703830878318114698103123328, −2.27443036991164184712954681481, −1.63109577663292307686131582235,
0.36108278711076275167114763181, 2.14522237272135846825053611742, 3.40464589794333083612689558943, 4.83806845669922344719147906274, 5.62562566114194258058017237292, 6.56788879718164349960790873758, 7.15345671957213740622297165524, 7.928559318477334005247373526463, 9.015555982403747470621258651825, 9.550939847471400667471613831832