L(s) = 1 | − 1.41i·2-s + (0.880 − 2.86i)3-s − 2.00·4-s + 2.23i·5-s + (−4.05 − 1.24i)6-s − 3.20·7-s + 2.82i·8-s + (−7.45 − 5.04i)9-s + 3.16·10-s + 14.6i·11-s + (−1.76 + 5.73i)12-s + 18.4·13-s + 4.53i·14-s + (6.41 + 1.96i)15-s + 4.00·16-s + 27.0i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.293 − 0.955i)3-s − 0.500·4-s + 0.447i·5-s + (−0.675 − 0.207i)6-s − 0.458·7-s + 0.353i·8-s + (−0.827 − 0.560i)9-s + 0.316·10-s + 1.33i·11-s + (−0.146 + 0.477i)12-s + 1.42·13-s + 0.324i·14-s + (0.427 + 0.131i)15-s + 0.250·16-s + 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.610131998\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.610131998\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-0.880 + 2.86i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + 4.79iT \) |
good | 7 | \( 1 + 3.20T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 - 18.4T + 169T^{2} \) |
| 17 | \( 1 - 27.0iT - 289T^{2} \) |
| 19 | \( 1 + 5.52T + 361T^{2} \) |
| 29 | \( 1 + 8.45iT - 841T^{2} \) |
| 31 | \( 1 - 49.6T + 961T^{2} \) |
| 37 | \( 1 - 42.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 17.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 52.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 3.72iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.31T + 3.72e3T^{2} \) |
| 67 | \( 1 - 0.327T + 4.48e3T^{2} \) |
| 71 | \( 1 + 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 22.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 68.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 72.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 89.0T + 9.40e3T^{2} \) |
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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
−10.29358222218680451636444203704, −9.450882714824862194699067802016, −8.423873108709377560605322941242, −7.79507858756216572213591409195, −6.50628902856849626326203749088, −6.11465846452268642656451145521, −4.39428984952113365869410604921, −3.38875556588473076816426858699, −2.28746356600113825664038036300, −1.24711355156841704535246616179,
0.63040296899054402283664669207, 2.96751128241928780725208239857, 3.82944765738678753676265489017, 4.91053904525633754491337270400, 5.77529110475259860920322425415, 6.58012015963676190855370581284, 8.006103844582549892301910135548, 8.609622048538265009520201586520, 9.258836228131114680322243301736, 10.07830903342316276995302680239