| L(s) = 1 | + (4.5 + 7.79i)3-s + (−11.0 + 19.1i)5-s + (−126. + 26.6i)7-s + (−40.5 + 70.1i)9-s + (208. + 360. i)11-s + 797.·13-s − 198.·15-s + (687. + 1.19e3i)17-s + (1.15e3 − 2.00e3i)19-s + (−778. − 869. i)21-s + (−477. + 827. i)23-s + (1.31e3 + 2.28e3i)25-s − 729·27-s − 7.03e3·29-s + (630. + 1.09e3i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.197 + 0.341i)5-s + (−0.978 + 0.205i)7-s + (−0.166 + 0.288i)9-s + (0.519 + 0.899i)11-s + 1.30·13-s − 0.227·15-s + (0.577 + 0.999i)17-s + (0.734 − 1.27i)19-s + (−0.385 − 0.430i)21-s + (−0.188 + 0.326i)23-s + (0.422 + 0.731i)25-s − 0.192·27-s − 1.55·29-s + (0.117 + 0.204i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.316998916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316998916\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (126. - 26.6i)T \) |
| good | 5 | \( 1 + (11.0 - 19.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-208. - 360. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 797.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-687. - 1.19e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.15e3 + 2.00e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (477. - 827. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 7.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-630. - 1.09e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (4.88e3 - 8.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 5.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.02e3 + 1.78e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (9.01e3 + 1.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.71e3 - 6.43e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.74e3 - 3.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-7.92e3 - 1.37e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.95e4 + 3.38e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.88e3 + 8.45e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 7.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (7.21e4 - 1.24e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 7.93e4T + 8.58e9T^{2} \) |
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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
−11.08292648495588681860366588438, −10.10197503684529526236258207510, −9.362013074810042908633863679676, −8.530678442921327649579601560416, −7.25176614841748258323871493723, −6.39871181117403138415471448182, −5.22573297389418479673743106928, −3.79375999447260143362188845650, −3.18652749002496854757248991718, −1.54572592599928544773981665960,
0.34035693491137297772179044406, 1.40741748889352541167801562551, 3.16981203884020678710840393660, 3.83235626691317496463558393072, 5.60952194428769375511132234178, 6.37797623537009223601277914757, 7.46045476203689725062209569663, 8.455958252020127116657901535127, 9.226383918285151176647271872339, 10.21634098439687661354715587828
