L(s) = 1 | + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (−4.23 − 1.13i)7-s + (−1.5 − 2.59i)9-s + 3.46i·12-s + (2.59 + 2.5i)13-s + (1.99 − 3.46i)16-s + (−0.830 + 3.09i)19-s + (5.36 − 5.36i)21-s + 5i·25-s + 5.19·27-s + (−8.46 + 2.26i)28-s + (0.830 + 0.830i)31-s + (−5.19 − 3i)36-s + (−3.09 − 11.5i)37-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (−1.59 − 0.428i)7-s + (−0.5 − 0.866i)9-s + 0.999i·12-s + (0.720 + 0.693i)13-s + (0.499 − 0.866i)16-s + (−0.190 + 0.710i)19-s + (1.17 − 1.17i)21-s + i·25-s + 1.00·27-s + (−1.59 + 0.428i)28-s + (0.149 + 0.149i)31-s + (−0.866 − 0.5i)36-s + (−0.509 − 1.90i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.706662 + 0.109269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.706662 + 0.109269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 1.5i)T \) |
| 13 | \( 1 + (-2.59 - 2.5i)T \) |
good | 2 | \( 1 + (-1.73 + i)T^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + (4.23 + 1.13i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.830 - 3.09i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.830 - 0.830i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.09 + 11.5i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (4.33 + 7.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-15.7 + 4.23i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (7.63 - 7.63i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (2.57 - 9.59i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−16.19415011714200628062315027242, −15.59384536352613973508526326816, −14.23104170099924442255296359959, −12.62207203507745843343992234168, −11.27827400375444776803415132348, −10.28492346517608797924035033166, −9.311099632916972730542005383458, −6.84682932914977747208997913860, −5.83187567293754489209147142198, −3.59850093997033233524973792826,
2.90998648777473488123802691863, 6.04231259098432226043157725372, 6.87588616744967647746205702468, 8.407971758694125939444127237067, 10.33799096166553589972307571081, 11.68233344115948147884910941145, 12.65208779821095528365292614123, 13.40704597303938166863939693732, 15.47068144434049292699101142043, 16.24644086000657269718930235816