L(s) = 1 | + (6.39 − 3.69i)5-s + (−3.39 + 5.88i)7-s + (−5.29 − 3.05i)11-s + (8.39 + 14.5i)13-s − 25.1i·17-s − 17.5·19-s + (12.3 − 7.15i)23-s + (14.7 − 25.6i)25-s + (16.1 + 9.35i)29-s + (−23.3 − 40.5i)31-s + 50.2i·35-s + 49.5·37-s + (34.5 − 19.9i)41-s + (22.0 − 38.2i)43-s + (28.8 + 16.6i)47-s + ⋯ |
L(s) = 1 | + (1.27 − 0.738i)5-s + (−0.485 + 0.841i)7-s + (−0.481 − 0.278i)11-s + (0.646 + 1.11i)13-s − 1.48i·17-s − 0.926·19-s + (0.539 − 0.311i)23-s + (0.591 − 1.02i)25-s + (0.558 + 0.322i)29-s + (−0.754 − 1.30i)31-s + 1.43i·35-s + 1.34·37-s + (0.841 − 0.485i)41-s + (0.513 − 0.890i)43-s + (0.612 + 0.353i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.375955434\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.375955434\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-6.39 + 3.69i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.39 - 5.88i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (5.29 + 3.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.39 - 14.5i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 25.1iT - 289T^{2} \) |
| 19 | \( 1 + 17.5T + 361T^{2} \) |
| 23 | \( 1 + (-12.3 + 7.15i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.1 - 9.35i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (23.3 + 40.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-22.0 + 38.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-28.8 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.2 + 8.25i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 18.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.4 - 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 30.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-55.7 + 96.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-85.0 - 49.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 75.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-70.2 + 121. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.224454548053187375963853369897, −8.611693480231682762190552936627, −7.41879478383430594777320018942, −6.37151299657640513641369047999, −5.86596597334672272893419842730, −5.09504457202764020781060325587, −4.17633513110979068804825731805, −2.68896821245819096831910894596, −2.09463164723989703947101496843, −0.71702980053561517363961142947,
1.04478869941113221053507293258, 2.24894719206321869406560771535, 3.17590317597672311108422397440, 4.10573286812878458859996317913, 5.35570672241337315871028624429, 6.15179433077466584368758205719, 6.60690204339630766507547119173, 7.61305629842975669307550116613, 8.398573045145746832308717464942, 9.409128055118938965738387392697