L(s) = 1 | + (1 + i)2-s + (1.86 − 2.35i)3-s + 2i·4-s + (−1.91 − 4.61i)5-s + (4.21 − 0.488i)6-s + (−3.12 − 6.26i)7-s + (−2 + 2i)8-s + (−2.05 − 8.76i)9-s + (2.70 − 6.53i)10-s + 0.117i·11-s + (4.70 + 3.72i)12-s + (9.72 − 9.72i)13-s + (3.13 − 9.39i)14-s + (−14.4 − 4.09i)15-s − 4·16-s + (−13.8 + 13.8i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.620 − 0.783i)3-s + 0.5i·4-s + (−0.383 − 0.923i)5-s + (0.702 − 0.0814i)6-s + (−0.446 − 0.894i)7-s + (−0.250 + 0.250i)8-s + (−0.228 − 0.973i)9-s + (0.270 − 0.653i)10-s + 0.0106i·11-s + (0.391 + 0.310i)12-s + (0.747 − 0.747i)13-s + (0.223 − 0.670i)14-s + (−0.961 − 0.273i)15-s − 0.250·16-s + (−0.815 + 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74483 - 1.13377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74483 - 1.13377i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (-1.86 + 2.35i)T \) |
| 5 | \( 1 + (1.91 + 4.61i)T \) |
| 7 | \( 1 + (3.12 + 6.26i)T \) |
good | 11 | \( 1 - 0.117iT - 121T^{2} \) |
| 13 | \( 1 + (-9.72 + 9.72i)T - 169iT^{2} \) |
| 17 | \( 1 + (13.8 - 13.8i)T - 289iT^{2} \) |
| 19 | \( 1 - 29.7T + 361T^{2} \) |
| 23 | \( 1 + (2.25 - 2.25i)T - 529iT^{2} \) |
| 29 | \( 1 - 46.0T + 841T^{2} \) |
| 31 | \( 1 - 1.50iT - 961T^{2} \) |
| 37 | \( 1 + (-5.32 + 5.32i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 13.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.8 - 36.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (29.7 - 29.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-59.8 + 59.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (34.4 - 34.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 77.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (41.3 - 41.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 0.865iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (99.0 + 99.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-15.7 - 15.7i)T + 9.40e3iT^{2} \) |
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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
−12.38366381407589812749604467475, −11.29095916298772717800267672541, −9.780064947840205548614270165835, −8.588605605076954408766283971206, −7.906901614981285128967997545757, −6.93223203884076269500488810063, −5.81503752910077923765507572468, −4.28997929506739774441827161225, −3.21683008583288331384873514386, −1.00475337420811099762516807647,
2.49858969174948579195954744641, 3.34460579354976568873641407872, 4.55874921491193358232792699218, 5.92108344007806780599128769978, 7.17244837669615848622650174125, 8.655189040995032693132984119310, 9.484095726080879454567706142040, 10.43498251791662452635568689569, 11.41260124625142090295430114133, 12.04546637083288962718170391367