L(s) = 1 | + (−3.63 + 1.67i)2-s − 5.19i·3-s + (10.3 − 12.1i)4-s − 41.3·5-s + (8.70 + 18.8i)6-s + 18.5i·7-s + (−17.3 + 61.5i)8-s − 27·9-s + (150. − 69.2i)10-s − 41.7i·11-s + (−63.2 − 53.9i)12-s + 260.·13-s + (−31.0 − 67.2i)14-s + 215. i·15-s + (−40.0 − 252. i)16-s + 208.·17-s + ⋯ |
L(s) = 1 | + (−0.908 + 0.418i)2-s − 0.577i·3-s + (0.649 − 0.760i)4-s − 1.65·5-s + (0.241 + 0.524i)6-s + 0.377i·7-s + (−0.271 + 0.962i)8-s − 0.333·9-s + (1.50 − 0.692i)10-s − 0.344i·11-s + (−0.439 − 0.374i)12-s + 1.54·13-s + (−0.158 − 0.343i)14-s + 0.955i·15-s + (−0.156 − 0.987i)16-s + 0.722·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.650185 + 0.299734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.650185 + 0.299734i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.63 - 1.67i)T \) |
| 3 | \( 1 + 5.19iT \) |
| 7 | \( 1 - 18.5iT \) |
good | 5 | \( 1 + 41.3T + 625T^{2} \) |
| 11 | \( 1 + 41.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 260.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 208.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 418. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 471. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 835.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 760. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 388.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.40e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.10e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 725. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.26e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 6.73e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.63e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.70e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.11e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.60e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.18e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.65e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.09e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.12e4T + 8.85e7T^{2} \) |
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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
−13.86854739360626196809869717901, −12.18758759275189392168718474324, −11.55238032730551893842580434193, −10.46812411707075405543770500911, −8.650773760361380375264342979407, −8.140280665897444674623834118288, −7.04762600819710439368004095234, −5.71354866068371302356866009744, −3.48933321793640190966056960640, −1.10073740889731933321843712866,
0.61473514635534073678139868083, 3.27012701082199429423008269370, 4.31650483429371171947536827027, 6.76810169506126570938593053783, 8.010406404086195337226649147300, 8.759894092530993091836235150184, 10.22722985975045946536407107582, 11.17473587985043880732216373077, 11.81995275481466707711879578012, 13.09335105380406054937060457433