L(s) = 1 | + 1.41·2-s − 3.16i·3-s + 2.00·4-s + 2.23i·5-s − 4.47i·6-s + (−2.12 + 1.58i)7-s + 2.82·8-s − 7.00·9-s + 3.16i·10-s − 6.32i·12-s + (−3 + 2.23i)14-s + 7.07·15-s + 4.00·16-s − 9.89·18-s + 4.47i·20-s + (5.00 + 6.70i)21-s + ⋯ |
L(s) = 1 | + 1.00·2-s − 1.82i·3-s + 1.00·4-s + 0.999i·5-s − 1.82i·6-s + (−0.801 + 0.597i)7-s + 1.00·8-s − 2.33·9-s + 1.00i·10-s − 1.82i·12-s + (−0.801 + 0.597i)14-s + 1.82·15-s + 1.00·16-s − 2.33·18-s + 1.00i·20-s + (1.09 + 1.46i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57771 - 0.791799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57771 - 0.791799i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (2.12 - 1.58i)T \) |
good | 3 | \( 1 + 3.16iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.11225239304732580402977566748, −12.11742453549534851931297917120, −11.61614010864774147896112068620, −10.26525886636280949643501151289, −8.350228900637723750843926058179, −7.13075495728044061005542215735, −6.55748667978388838877340787042, −5.67462609290882247091427490012, −3.22841497703215902875557833550, −2.18875343104242022248358488470,
3.25283978547436296107913775342, 4.29000442484728781500519357480, 5.08321858398493089826632180994, 6.30741294212657032672330689926, 8.192468120997586729076353333112, 9.522942152189979853494718947006, 10.23030811660613297164910181896, 11.26528155726475995419553296090, 12.33570525716381753223274872403, 13.43892351819162010004748611023