| L(s) = 1 | + (−2.73 − 0.732i)2-s + (8.28 + 4.78i)3-s + (6.92 + 4i)4-s + (−17.4 + 4.68i)5-s + (−19.1 − 19.1i)6-s + (−36.2 + 62.8i)7-s + (−15.9 − 16i)8-s + (5.29 + 9.16i)9-s + 51.1·10-s − 102. i·11-s + (38.2 + 66.3i)12-s + (−128. + 34.4i)13-s + (145. − 145. i)14-s + (−167. − 44.8i)15-s + (31.9 + 55.4i)16-s + (−36.3 + 135. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.920 + 0.531i)3-s + (0.433 + 0.250i)4-s + (−0.699 + 0.187i)5-s + (−0.531 − 0.531i)6-s + (−0.740 + 1.28i)7-s + (−0.249 − 0.250i)8-s + (0.0653 + 0.113i)9-s + 0.511·10-s − 0.845i·11-s + (0.265 + 0.460i)12-s + (−0.759 + 0.203i)13-s + (0.740 − 0.740i)14-s + (−0.743 − 0.199i)15-s + (0.124 + 0.216i)16-s + (−0.125 + 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.191936 + 0.653820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191936 + 0.653820i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 37 | \( 1 + (-664. + 1.19e3i)T \) |
| good | 3 | \( 1 + (-8.28 - 4.78i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (17.4 - 4.68i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (36.2 - 62.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 102. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (128. - 34.4i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (36.3 - 135. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (527. - 141. i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-686. - 686. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (892. - 892. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-439. + 439. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-876. - 506. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.23e3 - 1.23e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.11e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-995. - 1.72e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (392. - 1.46e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-341. - 1.27e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (3.49e3 + 2.01e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-1.68e3 + 2.92e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 1.54e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.38e3 - 639. i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (3.35e3 + 5.81e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.07e3 - 1.36e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-1.10e4 - 1.10e4i)T + 8.85e7iT^{2} \) |
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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
−14.78067033828388873905413886544, −13.03908341934171709891231292476, −11.95406760418492872071457032793, −10.83133347732216308202147167468, −9.317557686366168887529145003126, −8.915991038341741151171093574420, −7.67611339409614365048440915054, −6.00604030448522425807376288854, −3.70817863666384392293587664520, −2.58967405894814621154284330725,
0.37029548201847488259470485262, 2.51440395858308604189449141246, 4.33272311087225930267850528379, 6.86887155654232934139680273891, 7.50668060918966386091533506622, 8.593775208392976247892296458186, 9.826632737302750370802402928468, 10.90989309783631860334087030652, 12.49986936244291381015250253979, 13.35303553865366547676673503651
