| L(s) = 1 | + (−2.56 − 1.54i)3-s + (−1.25 + 0.725i)7-s + (4.20 + 7.95i)9-s + (8.07 − 4.66i)11-s + (−9.70 − 5.60i)13-s − 4.16·17-s − 3.14·19-s + (4.34 + 0.0803i)21-s + (−6.69 + 11.5i)23-s + (1.49 − 26.9i)27-s + (−12.6 + 7.33i)29-s + (7.07 − 12.2i)31-s + (−27.9 − 0.516i)33-s − 18.0i·37-s + (16.2 + 29.4i)39-s + ⋯ |
| L(s) = 1 | + (−0.856 − 0.515i)3-s + (−0.179 + 0.103i)7-s + (0.467 + 0.883i)9-s + (0.733 − 0.423i)11-s + (−0.746 − 0.430i)13-s − 0.244·17-s − 0.165·19-s + (0.207 + 0.00382i)21-s + (−0.291 + 0.504i)23-s + (0.0553 − 0.998i)27-s + (−0.437 + 0.252i)29-s + (0.228 − 0.395i)31-s + (−0.847 − 0.0156i)33-s − 0.487i·37-s + (0.417 + 0.754i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5729851315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5729851315\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.56 + 1.54i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.25 - 0.725i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.07 + 4.66i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (9.70 + 5.60i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.16T + 289T^{2} \) |
| 19 | \( 1 + 3.14T + 361T^{2} \) |
| 23 | \( 1 + (6.69 - 11.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (12.6 - 7.33i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-7.07 + 12.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 18.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-17.6 - 10.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (28.9 - 16.7i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-0.946 - 1.63i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 97.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-28.9 - 16.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-29.6 - 51.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-83.0 - 47.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-66.7 - 115. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (3.74 + 6.48i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 100. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-56.1 + 32.4i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.14735487069875199250207589774, −9.443236604047641017987629988235, −8.317512549555396096365334678252, −7.47459812925633245645543165605, −6.63482676631852873598213723360, −5.87117396495559236789673429836, −5.01606939049019866439389766119, −3.91087220855831025058361174670, −2.49439617147166438953967363575, −1.16151791666920264132248449777,
0.23080572780739955508790546101, 1.85578619123003095173579807023, 3.47173182916479793747380984274, 4.45066691078373799529062557110, 5.14136657227920752087576075728, 6.39663885549964150458962950657, 6.79162633069318182374581518737, 7.971346305501081914740547956085, 9.198382511780980596619727908203, 9.686578874790453844338907662613
