L(s) = 1 | + (3.32 − 1.91i)5-s + (−2.70 + 4.68i)7-s + (10.4 + 6.01i)11-s + (0.140 + 0.242i)13-s − 22.7i·17-s + 2.05·19-s + (26.8 − 15.5i)23-s + (−5.13 + 8.90i)25-s + (−4.59 − 2.65i)29-s + (9.81 + 17.0i)31-s + 20.7i·35-s − 65.7·37-s + (59.1 − 34.1i)41-s + (28.1 − 48.7i)43-s + (3.83 + 2.21i)47-s + ⋯ |
L(s) = 1 | + (0.664 − 0.383i)5-s + (−0.386 + 0.669i)7-s + (0.946 + 0.546i)11-s + (0.0107 + 0.0186i)13-s − 1.33i·17-s + 0.108·19-s + (1.16 − 0.674i)23-s + (−0.205 + 0.356i)25-s + (−0.158 − 0.0914i)29-s + (0.316 + 0.548i)31-s + 0.593i·35-s − 1.77·37-s + (1.44 − 0.832i)41-s + (0.653 − 1.13i)43-s + (0.0816 + 0.0471i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.396944654\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.396944654\) |
\(L(2)\) |
|
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 | \( 1 + (-3.32 + 1.91i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (2.70 - 4.68i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.4 - 6.01i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-0.140 - 0.242i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 22.7iT - 289T^{2} \) |
| 19 | \( 1 - 2.05T + 361T^{2} \) |
| 23 | \( 1 + (-26.8 + 15.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (4.59 + 2.65i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-9.81 - 17.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 65.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-59.1 + 34.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-28.1 + 48.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-3.83 - 2.21i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 34.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-23.4 + 13.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (51.7 - 89.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.3 - 68.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 94.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 126.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-45.2 + 78.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-45.1 - 26.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 72.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (9.38 - 16.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.083593925450747474407177107124, −8.762235673481018626381800059892, −7.30578306267673915069983171748, −6.84507297950566010858793834588, −5.78185280031056980689718883422, −5.18416088143075913142846165359, −4.20859114739143936876911183842, −3.01095865362444574600061981596, −2.08322713642387082882208336456, −0.877434763374457683336608573817,
0.884296461151362242058841717512, 1.98600968888808472891533551923, 3.30226676442444843803125173950, 3.92037090810657167411277686248, 5.09857455059982140850049407504, 6.23075028423287018529733653921, 6.47015665288375513690164485845, 7.50807575338312565696353044645, 8.367615661527928887215338776652, 9.317602909849730116007118143428