L(s) = 1 | + 2.22i·2-s + 2.89·3-s + 3.05·4-s + 19.1i·5-s + 6.44i·6-s − 4.29i·7-s + 24.5i·8-s − 18.6·9-s − 42.5·10-s − 11i·11-s + 8.86·12-s + (35.7 − 30.2i)13-s + 9.55·14-s + 55.4i·15-s − 30.1·16-s − 7.67·17-s + ⋯ |
L(s) = 1 | + 0.786i·2-s + 0.557·3-s + 0.382·4-s + 1.71i·5-s + 0.438i·6-s − 0.231i·7-s + 1.08i·8-s − 0.688·9-s − 1.34·10-s − 0.301i·11-s + 0.213·12-s + (0.763 − 0.645i)13-s + 0.182·14-s + 0.954i·15-s − 0.471·16-s − 0.109·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.892898 + 1.92391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.892898 + 1.92391i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + 11iT \) |
| 13 | \( 1 + (-35.7 + 30.2i)T \) |
good | 2 | \( 1 - 2.22iT - 8T^{2} \) |
| 3 | \( 1 - 2.89T + 27T^{2} \) |
| 5 | \( 1 - 19.1iT - 125T^{2} \) |
| 7 | \( 1 + 4.29iT - 343T^{2} \) |
| 17 | \( 1 + 7.67T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 15.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 238.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 4.36iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 348. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 175.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 277. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 714.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 831. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 774.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 67.2iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 534. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 433. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 100. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 63.8iT - 9.12e5T^{2} \) |
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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
−13.51707341097612536476251369782, −11.69394477975561433487499152894, −10.96993449660497860638523499264, −10.07908130542375823319959012516, −8.355811809315454338023342991723, −7.70175941264856999362433020700, −6.51061846385080715747794996412, −5.83785658509240032309828912520, −3.47366882497540026997911857960, −2.49135554064643747239131590840,
1.01945339496615248918280764525, 2.44246325364849127785773655922, 4.00606449781187456890550939951, 5.35664273154730005858687493336, 6.92469509516899385824175029116, 8.704254444678512918264895577984, 8.837976433896238096745835580808, 10.23463693993474765684138998353, 11.56261355450461312088315232807, 12.14347586042034618239771621913