L(s) = 1 | + (1.86 + 1.56i)3-s + (−2.06 + 0.752i)5-s + (−1.48 − 2.57i)7-s + (0.509 + 2.89i)9-s + (1.34 − 2.33i)11-s + (3.64 − 3.05i)13-s + (−5.03 − 1.83i)15-s + (−1.19 + 6.79i)17-s + (−4.33 − 0.477i)19-s + (1.25 − 7.12i)21-s + (−4.86 − 1.77i)23-s + (−0.121 + 0.101i)25-s + (0.0792 − 0.137i)27-s + (1.17 + 6.63i)29-s + (2.14 + 3.71i)31-s + ⋯ |
L(s) = 1 | + (1.07 + 0.904i)3-s + (−0.924 + 0.336i)5-s + (−0.560 − 0.971i)7-s + (0.169 + 0.963i)9-s + (0.406 − 0.704i)11-s + (1.01 − 0.848i)13-s + (−1.30 − 0.473i)15-s + (−0.290 + 1.64i)17-s + (−0.993 − 0.109i)19-s + (0.274 − 1.55i)21-s + (−1.01 − 0.369i)23-s + (−0.0242 + 0.0203i)25-s + (0.0152 − 0.0264i)27-s + (0.217 + 1.23i)29-s + (0.384 + 0.666i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03771 + 0.305438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03771 + 0.305438i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (4.33 + 0.477i)T \) |
good | 3 | \( 1 + (-1.86 - 1.56i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (2.06 - 0.752i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.48 + 2.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.34 + 2.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.64 + 3.05i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.19 - 6.79i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (4.86 + 1.77i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.17 - 6.63i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 + (-1.42 - 1.19i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.50 + 3.09i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.108 - 0.617i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (6.42 + 2.33i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.623 + 3.53i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.43 - 3.43i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.58 + 8.96i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.533 - 0.194i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.598 + 0.502i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.42 - 2.03i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.64 - 6.31i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.84 + 6.58i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.47 - 8.34i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.70403890910427482753776569486, −13.79177575048590821064396734061, −12.63493515030478167611413337633, −10.84452998669670605617900682655, −10.36594224214373633636747335916, −8.758721796620368334394631341730, −8.063235382847073726327715719598, −6.42424493897741305481972712634, −3.97100413618644408948050007232, −3.49696179161483934918036708375,
2.35201575711580772377083481448, 4.11255174678781203528980719732, 6.37460372075490149179782892675, 7.61328810229262724220872388146, 8.641390070452511158730736387942, 9.456277647325743244928594175813, 11.60232938896238708590829814521, 12.28569148126337437649122520299, 13.37325128919212044543280127446, 14.27519022709450320365428683077