L(s) = 1 | + (5.59 + 3.22i)3-s + (−4.68 + 8.12i)5-s − 33.5·7-s + (−19.6 − 34.0i)9-s + 58.1·11-s + (−29.5 + 17.0i)13-s + (−52.4 + 30.2i)15-s + (−9.99 + 17.3i)17-s + (115. − 342. i)19-s + (−187. − 108. i)21-s + (−459. − 796. i)23-s + (268. + 465. i)25-s − 776. i·27-s + (685. − 395. i)29-s − 517. i·31-s + ⋯ |
L(s) = 1 | + (0.621 + 0.358i)3-s + (−0.187 + 0.324i)5-s − 0.684·7-s + (−0.242 − 0.420i)9-s + 0.480·11-s + (−0.175 + 0.101i)13-s + (−0.233 + 0.134i)15-s + (−0.0345 + 0.0598i)17-s + (0.318 − 0.947i)19-s + (−0.425 − 0.245i)21-s + (−0.869 − 1.50i)23-s + (0.429 + 0.744i)25-s − 1.06i·27-s + (0.815 − 0.470i)29-s − 0.538i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.528703891\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.528703891\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-115. + 342. i)T \) |
good | 3 | \( 1 + (-5.59 - 3.22i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (4.68 - 8.12i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + 33.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 58.1T + 1.46e4T^{2} \) |
| 13 | \( 1 + (29.5 - 17.0i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (9.99 - 17.3i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 23 | \( 1 + (459. + 796. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-685. + 395. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + 517. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 181. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-738. - 426. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.24e3 + 2.14e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (158. + 274. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.24e3 - 719. i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (751. + 433. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (403. + 699. i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-5.97e3 + 3.44e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (617. + 356. i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-3.60e3 + 6.23e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.64e3 + 2.10e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.98e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-1.56e3 + 902. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (599. + 346. i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−10.81157042699694225766405678516, −9.763050278277329614789951179236, −9.117871088899010109477503625235, −8.174450723903694945697081673345, −6.90599704469781855977216431801, −6.12930208165842764679956342368, −4.52268056103582330128903289926, −3.46449888577719398242957941971, −2.50625902837404989539352689362, −0.44432563401834314119089551337,
1.35387635038162836399759665815, 2.77225530500086256830378165732, 3.87613252728632722252056077897, 5.29583533610832251651523320137, 6.44872547194185505001971985564, 7.59642787582826411352332706233, 8.334866301224955429099044633555, 9.327330873277564537391188525781, 10.17076445215107925087771352780, 11.37414844036890360083823951251