L(s) = 1 | + (−0.722 − 1.21i)2-s + (0.294 + 1.70i)3-s + (−0.956 + 1.75i)4-s + (3.17 − 1.83i)5-s + (1.86 − 1.59i)6-s + (−0.191 + 0.332i)7-s + (2.82 − 0.104i)8-s + (−2.82 + 1.00i)9-s + (−4.51 − 2.53i)10-s + (−1.73 − 1.00i)11-s + (−3.27 − 1.11i)12-s + (−0.397 + 0.229i)13-s + (0.542 − 0.00670i)14-s + (4.06 + 4.87i)15-s + (−2.16 − 3.36i)16-s − 4.08·17-s + ⋯ |
L(s) = 1 | + (−0.510 − 0.859i)2-s + (0.170 + 0.985i)3-s + (−0.478 + 0.878i)4-s + (1.41 − 0.819i)5-s + (0.760 − 0.649i)6-s + (−0.0725 + 0.125i)7-s + (0.999 − 0.0370i)8-s + (−0.942 + 0.335i)9-s + (−1.42 − 0.801i)10-s + (−0.524 − 0.302i)11-s + (−0.946 − 0.322i)12-s + (−0.110 + 0.0636i)13-s + (0.145 − 0.00179i)14-s + (1.04 + 1.25i)15-s + (−0.542 − 0.840i)16-s − 0.990·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.824570 - 0.132774i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.824570 - 0.132774i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.722 + 1.21i)T \) |
| 3 | \( 1 + (-0.294 - 1.70i)T \) |
good | 5 | \( 1 + (-3.17 + 1.83i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.191 - 0.332i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.397 - 0.229i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 4.72iT - 19T^{2} \) |
| 23 | \( 1 + (2.97 + 5.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.03 + 1.17i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.592 - 1.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.74iT - 37T^{2} \) |
| 41 | \( 1 + (-4.75 - 8.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.03 - 0.598i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.27 + 5.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.63iT - 53T^{2} \) |
| 59 | \( 1 + (-0.603 + 0.348i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.23 - 2.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.87 + 5.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + (5.35 - 9.28i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.49 + 3.16i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 + (2.98 - 5.17i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.31770957606260732267827940488, −13.40805502865013923296912720633, −12.40212873282210106012459199376, −10.89057365409888261801898347695, −10.01750106585734601589370179042, −9.197450160623024686250982239053, −8.308947970721994279573584367014, −5.76145161413131067301617516217, −4.38263627593294020065297729947, −2.38219152355092295704020305647,
2.20081231374687686700906514321, 5.49849021347436200021081880632, 6.57222035757360172631357160673, 7.39826617377630839106589114942, 8.909328883555814232249004502370, 9.958635252627121765068837939167, 11.13767299276315454793937979875, 13.14780265970019701808722195196, 13.66129853639989723529139694620, 14.58954950582783175156433846057