L(s) = 1 | + (−1.61 − 1.18i)2-s + 3.23i·3-s + (1.18 + 3.81i)4-s + (−3.82 − 3.21i)5-s + (3.83 − 5.20i)6-s + (6.97 − 0.588i)7-s + (2.61 − 7.56i)8-s − 1.44·9-s + (2.34 + 9.72i)10-s + 13.5i·11-s + (−12.3 + 3.83i)12-s − 21.1i·13-s + (−11.9 − 7.32i)14-s + (10.3 − 12.3i)15-s + (−13.1 + 9.07i)16-s + 0.174·17-s + ⋯ |
L(s) = 1 | + (−0.805 − 0.592i)2-s + 1.07i·3-s + (0.296 + 0.954i)4-s + (−0.765 − 0.643i)5-s + (0.638 − 0.867i)6-s + (0.996 − 0.0840i)7-s + (0.327 − 0.945i)8-s − 0.160·9-s + (0.234 + 0.972i)10-s + 1.23i·11-s + (−1.02 + 0.319i)12-s − 1.63i·13-s + (−0.852 − 0.523i)14-s + (0.693 − 0.824i)15-s + (−0.823 + 0.567i)16-s + 0.0102·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02032 + 0.328337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02032 + 0.328337i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.61 + 1.18i)T \) |
| 5 | \( 1 + (3.82 + 3.21i)T \) |
| 7 | \( 1 + (-6.97 + 0.588i)T \) |
good | 3 | \( 1 - 3.23iT - 9T^{2} \) |
| 11 | \( 1 - 13.5iT - 121T^{2} \) |
| 13 | \( 1 + 21.1iT - 169T^{2} \) |
| 17 | \( 1 - 0.174T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 - 27.0iT - 529T^{2} \) |
| 29 | \( 1 - 18.8iT - 841T^{2} \) |
| 31 | \( 1 - 7.48iT - 961T^{2} \) |
| 37 | \( 1 - 66.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.05T + 2.20e3T^{2} \) |
| 53 | \( 1 + 60.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 14.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 35.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 66.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 51.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 77.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 94.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 59.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 28.6T + 9.40e3T^{2} \) |
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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
−11.47453943597100640990018044826, −10.72416634722972037412070951885, −9.821106872246246944395725863530, −9.141576213515054040794396117343, −7.83866986564890237963901486241, −7.55884132408657007094123675714, −5.18282337326576273989028791096, −4.38114302321231379948007801079, −3.24144909341283583641816954444, −1.24502868153281265494534473224,
0.871451901102943890856725661182, 2.32292168060215952233362182383, 4.40102534065926523485435240352, 6.02142751205550720666621698418, 6.84381318614773628604621869334, 7.70680218399374741182899581251, 8.281593425585146057185952809504, 9.372750072428485115546850335185, 10.84033639697842866574750697903, 11.42168586478016839017416922032