| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (4.60 − 1.95i)5-s + 2.44·6-s + (−2.49 + 6.54i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (0.985 + 7.00i)10-s + (1.59 − 2.75i)11-s + (−0.896 + 3.34i)12-s + (16.7 − 16.7i)13-s + (−8.02 − 5.79i)14-s + (−5.33 − 6.82i)15-s + (1.99 + 3.46i)16-s + (14.5 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.920 − 0.390i)5-s + 0.408·6-s + (−0.356 + 0.934i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.0985 + 0.700i)10-s + (0.144 − 0.250i)11-s + (−0.0747 + 0.278i)12-s + (1.28 − 1.28i)13-s + (−0.573 − 0.414i)14-s + (−0.355 − 0.454i)15-s + (0.124 + 0.216i)16-s + (0.855 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00411i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50788 + 0.00310253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50788 + 0.00310253i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + (0.448 + 1.67i)T \) |
| 5 | \( 1 + (-4.60 + 1.95i)T \) |
| 7 | \( 1 + (2.49 - 6.54i)T \) |
| good | 11 | \( 1 + (-1.59 + 2.75i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-16.7 + 16.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (-14.5 + 3.89i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-13.5 + 7.79i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-5.41 - 1.45i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 47.4iT - 841T^{2} \) |
| 31 | \( 1 + (-0.731 + 1.26i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-14.3 + 53.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 27.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (16.8 - 16.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (7.04 - 26.2i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (8.92 + 33.3i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-46.3 - 26.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.4 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-59.1 + 15.8i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 102.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (7.92 + 29.5i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (101. - 58.3i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (74.9 - 74.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (91.7 - 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (66.3 + 66.3i)T + 9.40e3iT^{2} \) |
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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
−12.50873738559104586791054681207, −11.14666348646275332714614391950, −9.960555536984750472200220638384, −8.961170763225052391823577737709, −8.252319001677237734529438320499, −6.90536126819288271616475800071, −5.74350253361419725773779184166, −5.41120021399926172709430665947, −3.07960329783783520422468180573, −1.15314953742729619255252851704,
1.45379186528364725593391675206, 3.28052050858269486008194478015, 4.32507843453356389984083910442, 5.86693560900204703645893985506, 6.92514009228232698050266870886, 8.444393849913060412729218574653, 9.727139084172103137112994741191, 9.996201108891073387624340092968, 11.07375275985487566641741749750, 11.84598240302015362196186801874
