L(s) = 1 | + (−1.25 − 0.643i)2-s + (1.48 − 0.887i)3-s + (1.17 + 1.62i)4-s + (2.05 + 0.887i)5-s + (−2.44 + 0.161i)6-s + 7-s + (−0.434 − 2.79i)8-s + (1.42 − 2.64i)9-s + (−2.01 − 2.43i)10-s − 2.75·11-s + (3.18 + 1.36i)12-s − 1.90i·13-s + (−1.25 − 0.643i)14-s + (3.84 − 0.501i)15-s + (−1.25 + 3.79i)16-s + 5.03·17-s + ⋯ |
L(s) = 1 | + (−0.890 − 0.454i)2-s + (0.858 − 0.512i)3-s + (0.586 + 0.810i)4-s + (0.917 + 0.397i)5-s + (−0.997 + 0.0659i)6-s + 0.377·7-s + (−0.153 − 0.988i)8-s + (0.474 − 0.880i)9-s + (−0.636 − 0.771i)10-s − 0.832·11-s + (0.918 + 0.395i)12-s − 0.529i·13-s + (−0.336 − 0.171i)14-s + (0.991 − 0.129i)15-s + (−0.312 + 0.949i)16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34282 - 0.579256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34282 - 0.579256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.25 + 0.643i)T \) |
| 3 | \( 1 + (-1.48 + 0.887i)T \) |
| 5 | \( 1 + (-2.05 - 0.887i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 + 1.90iT - 13T^{2} \) |
| 17 | \( 1 - 5.03T + 17T^{2} \) |
| 19 | \( 1 - 1.90iT - 19T^{2} \) |
| 23 | \( 1 - 2.05iT - 23T^{2} \) |
| 29 | \( 1 - 4.94iT - 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 6.48iT - 37T^{2} \) |
| 41 | \( 1 - 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 7.51iT - 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 + 0.716T + 59T^{2} \) |
| 61 | \( 1 - 5.13T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 + 6.11iT - 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 7.71iT - 83T^{2} \) |
| 89 | \( 1 + 13.5iT - 89T^{2} \) |
| 97 | \( 1 - 16.6iT - 97T^{2} \) |
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
−10.81831262022954566116884020286, −9.981542326120693729907951262414, −9.436864658475906478940434709554, −8.226987618297485661206273052413, −7.73899023622497509495261068176, −6.71706473704511560691514756878, −5.50224219398349956677962402680, −3.49817969393153026247029901432, −2.56609091831912134913664950147, −1.43388743619571612527640334698,
1.64233162408795059754949117661, 2.82791752532078518204897285208, 4.75633382747856799310152514561, 5.54584039397871492880115573703, 6.85753944160407103010087042915, 7.962542649945003405058540389834, 8.587444053244230099886978637612, 9.474286110766326823354685358863, 10.13081666180448902134580551883, 10.79066945250143758343815975863