| L(s) = 1 | + (−2.35 + 4.08i)2-s + (−14.3 − 24.9i)3-s + (4.88 + 8.45i)4-s + (−24.5 − 42.5i)5-s + 135.·6-s + (109. + 188. i)7-s − 196.·8-s + (−292. + 506. i)9-s + 231.·10-s + 122.·11-s + (140. − 243. i)12-s + (72.0 + 124. i)13-s − 1.02e3·14-s + (−707. + 1.22e3i)15-s + (308. − 533. i)16-s + (−810. + 1.40e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.416 + 0.721i)2-s + (−0.923 − 1.59i)3-s + (0.152 + 0.264i)4-s + (−0.439 − 0.761i)5-s + 1.53·6-s + (0.840 + 1.45i)7-s − 1.08·8-s + (−1.20 + 2.08i)9-s + 0.732·10-s + 0.304·11-s + (0.281 − 0.488i)12-s + (0.118 + 0.204i)13-s − 1.40·14-s + (−0.811 + 1.40i)15-s + (0.300 − 0.521i)16-s + (−0.679 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.322347 + 0.466316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322347 + 0.466316i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (6.16e3 - 5.60e3i)T \) |
| good | 2 | \( 1 + (2.35 - 4.08i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (14.3 + 24.9i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (24.5 + 42.5i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-109. - 188. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 - 122.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-72.0 - 124. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (810. - 1.40e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-917. - 1.58e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + 1.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 376.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.69e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-4.29e3 - 7.44e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.25e4 + 2.17e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.94e3 - 5.09e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.48e4 - 4.30e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.83e4 + 3.18e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.58e4 - 4.47e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 - 5.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.37e4 + 2.38e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.70e4 - 6.42e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.20e4 + 5.55e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 8.13e4T + 8.58e9T^{2} \) |
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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
−16.11465835001695712676868986737, −14.73036305652402821117845901248, −12.88505175740468794435266878974, −12.06532239135923703153146342433, −11.53748406857224177328838894165, −8.544290858187308517228371800954, −8.050490115983357324328405540841, −6.53176943358439709573203002863, −5.47182334338120209728395720943, −1.77464117429690930188586198990,
0.41726758433174985702502788978, 3.55903398408887655707045245242, 5.00968872286648934807101167717, 6.91566145075558203895367177292, 9.299155622176032916331537009312, 10.41350189056590255451543824474, 11.10430694613004188194329378088, 11.56375497348396781244484044513, 14.17991999687889574407033078166, 15.18743863293663864796862293241
