L(s) = 1 | + (−0.366 + 1.36i)2-s + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (0.678 + 4.95i)5-s − 2.44·6-s + (5.93 + 3.71i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (−7.01 − 0.886i)10-s + (−3.58 + 6.20i)11-s + (0.896 − 3.34i)12-s + (7.28 − 7.28i)13-s + (−7.24 + 6.74i)14-s + (−7.98 + 3.35i)15-s + (1.99 + 3.46i)16-s + (−18.1 + 4.87i)17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.135 + 0.990i)5-s − 0.408·6-s + (0.847 + 0.530i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.701 − 0.0886i)10-s + (−0.325 + 0.564i)11-s + (0.0747 − 0.278i)12-s + (0.560 − 0.560i)13-s + (−0.517 + 0.481i)14-s + (−0.532 + 0.223i)15-s + (0.124 + 0.216i)16-s + (−1.06 + 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.309296 + 1.33124i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.309296 + 1.33124i\) |
\(L(2)\) |
|
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 | \( 1 + (-0.678 - 4.95i)T \) |
| 7 | \( 1 + (-5.93 - 3.71i)T \) |
good | 11 | \( 1 + (3.58 - 6.20i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.28 + 7.28i)T - 169iT^{2} \) |
| 17 | \( 1 + (18.1 - 4.87i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (7.65 - 4.41i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-0.401 - 0.107i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 27.7iT - 841T^{2} \) |
| 31 | \( 1 + (-12.5 + 21.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-13.8 + 51.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 46.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (37.3 - 37.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-8.32 + 31.0i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-18.4 - 68.8i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-35.2 - 20.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-3.30 - 5.72i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-128. + 34.4i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 129.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-2.10 - 7.85i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-59.9 + 34.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-57.0 + 57.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (114. - 65.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (46.0 + 46.0i)T + 9.40e3iT^{2} \) |
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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
−12.67623407945049500251645221635, −11.18392149639196311061838100799, −10.65693947356668114974666196152, −9.535793053117276117298906244135, −8.497396111448581029377415364005, −7.60235802015414069096522306133, −6.37440097436623355816701090203, −5.32880298919208214826090968027, −4.05011156961553600062886998711, −2.34111054375010135518648298756,
0.833836080486318294907237642833, 2.15764996495389873856212065434, 4.04932976888020258338414522213, 5.11105170244594485522627691913, 6.64432094642354228208961724467, 8.144426297254965293928169698941, 8.572687977583754542815058579580, 9.719141348772686390283992485417, 11.04990137218754295309357707246, 11.59873591552447112665432000227