L(s) = 1 | + (−0.220 + 2.81i)2-s + (2.12 − 2.12i)3-s + (−7.90 − 1.24i)4-s + (10.2 + 10.2i)5-s + (5.51 + 6.44i)6-s + 32.8i·7-s + (5.23 − 22.0i)8-s − 8.99i·9-s + (−31.2 + 26.7i)10-s + (−18.2 − 18.2i)11-s + (−19.3 + 14.1i)12-s + (22.5 − 22.5i)13-s + (−92.5 − 7.22i)14-s + 43.6·15-s + (60.9 + 19.6i)16-s + 50.1·17-s + ⋯ |
L(s) = 1 | + (−0.0778 + 0.996i)2-s + (0.408 − 0.408i)3-s + (−0.987 − 0.155i)4-s + (0.920 + 0.920i)5-s + (0.375 + 0.438i)6-s + 1.77i·7-s + (0.231 − 0.972i)8-s − 0.333i·9-s + (−0.989 + 0.846i)10-s + (−0.499 − 0.499i)11-s + (−0.466 + 0.339i)12-s + (0.481 − 0.481i)13-s + (−1.76 − 0.137i)14-s + 0.751·15-s + (0.951 + 0.306i)16-s + 0.715·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0810 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0810 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.999166 + 1.08372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999166 + 1.08372i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.220 - 2.81i)T \) |
| 3 | \( 1 + (-2.12 + 2.12i)T \) |
good | 5 | \( 1 + (-10.2 - 10.2i)T + 125iT^{2} \) |
| 7 | \( 1 - 32.8iT - 343T^{2} \) |
| 11 | \( 1 + (18.2 + 18.2i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-22.5 + 22.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 50.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + (6.68 - 6.68i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 186. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-118. + 118. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-198. - 198. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 186. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-10.9 - 10.9i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 23.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-134. - 134. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-220. - 220. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (453. - 453. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (184. - 184. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 18.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 828. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (173. - 173. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 335. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 687.T + 9.12e5T^{2} \) |
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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
−15.20531585820277601577129795084, −14.56025090402860941377026957795, −13.48548714863013945609730508704, −12.36507137603392195714269774051, −10.40144396598291490132292418731, −9.083069026690404763087277667031, −8.087954917014549783947819040509, −6.37598493917465866386744516693, −5.62161551409682849707161192927, −2.71467913881193895021311253275,
1.41694261575109380428259167911, 3.78709459565377448655179353646, 5.12004266927910963320030664770, 7.69014491062083425677464836750, 9.245572766124051035240727178507, 10.01556659496407529057628612427, 11.05468715295429046018661938776, 12.82624265311239061564131100015, 13.53080455617173936553280667460, 14.33151620954949190536614801425