| L(s) = 1 | + (−0.987 + 0.569i)2-s + (−0.202 − 2.99i)3-s + (−1.35 + 2.33i)4-s + (−1.93 + 1.11i)5-s + (1.90 + 2.83i)6-s + (2.61 − 6.49i)7-s − 7.63i·8-s + (−8.91 + 1.21i)9-s + (1.27 − 2.20i)10-s + (−12.8 − 7.41i)11-s + (7.27 + 3.56i)12-s − 23.9·13-s + (1.12 + 7.89i)14-s + (3.73 + 5.56i)15-s + (−1.04 − 1.81i)16-s + (5.54 + 3.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.493 + 0.284i)2-s + (−0.0674 − 0.997i)3-s + (−0.337 + 0.584i)4-s + (−0.387 + 0.223i)5-s + (0.317 + 0.473i)6-s + (0.372 − 0.927i)7-s − 0.954i·8-s + (−0.990 + 0.134i)9-s + (0.127 − 0.220i)10-s + (−1.16 − 0.673i)11-s + (0.606 + 0.297i)12-s − 1.84·13-s + (0.0803 + 0.564i)14-s + (0.249 + 0.371i)15-s + (−0.0655 − 0.113i)16-s + (0.326 + 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.125307 - 0.360428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.125307 - 0.360428i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.202 + 2.99i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 7 | \( 1 + (-2.61 + 6.49i)T \) |
| good | 2 | \( 1 + (0.987 - 0.569i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (12.8 + 7.41i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 23.9T + 169T^{2} \) |
| 17 | \( 1 + (-5.54 - 3.20i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.55 - 6.16i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-28.8 + 16.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 32.6iT - 841T^{2} \) |
| 31 | \( 1 + (1.00 - 1.73i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-9.06 - 15.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 40.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-6.22 + 3.59i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (34.1 + 19.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (66.6 + 38.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 24.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-50.9 + 88.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 65.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-58.9 + 102. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.0 - 29.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 34.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (28.7 - 16.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 5.32T + 9.40e3T^{2} \) |
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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
−13.06010906963502419685878063417, −12.24959337923948627531222422357, −11.05538416024671611043542587178, −9.814310006311564268789414936398, −8.156248502557994654567458139352, −7.73230839241965739739210105873, −6.79521830875615764359924061358, −4.87517349067993007650083601252, −2.97082726165237751777158084461, −0.32336140297977028702953172164,
2.59615529657076163736618621786, 4.99515486640588147194497981606, 5.20370793727779653310592325599, 7.61415663575386098893808598711, 8.916766028425300903731014892732, 9.633239696165077932202387464353, 10.57375785522631412141902862308, 11.59035022039400677218191687289, 12.64232974518811013877444100815, 14.32126278104311709764796487755
