| L(s) = 1 | + (−3.43 + 7.22i)2-s + (−40.3 − 49.6i)4-s − 55.9·5-s + 472. i·7-s + (497. − 120. i)8-s + (192. − 403. i)10-s − 939. i·11-s + 22.5·13-s + (−3.40e3 − 1.62e3i)14-s + (−839. + 4.00e3i)16-s − 6.24e3·17-s + 1.11e4i·19-s + (2.25e3 + 2.77e3i)20-s + (6.78e3 + 3.23e3i)22-s + 1.16e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.429 + 0.902i)2-s + (−0.630 − 0.776i)4-s − 0.447·5-s + 1.37i·7-s + (0.971 − 0.235i)8-s + (0.192 − 0.403i)10-s − 0.705i·11-s + 0.0102·13-s + (−1.24 − 0.591i)14-s + (−0.205 + 0.978i)16-s − 1.27·17-s + 1.62i·19-s + (0.281 + 0.347i)20-s + (0.637 + 0.303i)22-s + 0.0960i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3656016983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3656016983\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.43 - 7.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 55.9T \) |
| good | 7 | \( 1 - 472. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 939. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 22.5T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.24e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.11e4iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.16e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.56e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.30e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.06e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.13e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.13e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.84e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.33e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 5.52e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.04e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.44e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.42e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.78e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.58e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 8.20e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.00e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.44e6T + 8.32e11T^{2} \) |
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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
−11.37365435631776825017394307671, −10.25870538939042213259708839923, −8.938137647142670313765199134649, −8.543193444312620450782356452644, −7.34635175431659836003656654538, −6.08784415615783368781238671875, −5.35728003717597590779508759746, −3.83832833255452083932068937602, −1.97939157831023620546153487287, −0.14486927022774846394052201737,
0.990999642317884579959549750430, 2.52124834942112688375216245065, 3.99620580186261379591223223186, 4.66837150750630087869157144364, 6.92902116497908783839567981507, 7.61218634432966432490610109475, 8.892431925568980946077667173948, 9.799051203535193492600748628553, 10.95693508264523984631401458357, 11.28254847539002447570795113457
