| L(s) = 1 | + (−0.285 + 0.493i)3-s + (12.7 + 47.3i)7-s + (40.3 + 69.8i)9-s + (−89.1 + 154. i)11-s − 116.·13-s + (−182. + 316. i)17-s + (98.9 − 57.1i)19-s + (−26.9 − 7.21i)21-s + (−38.6 + 22.3i)23-s − 92.1·27-s − 1.12e3·29-s + (−9.18 − 5.30i)31-s + (−50.8 − 88.1i)33-s + (1.30e3 − 756. i)37-s + (33.1 − 57.3i)39-s + ⋯ |
| L(s) = 1 | + (−0.0316 + 0.0548i)3-s + (0.259 + 0.965i)7-s + (0.497 + 0.862i)9-s + (−0.737 + 1.27i)11-s − 0.687·13-s + (−0.631 + 1.09i)17-s + (0.274 − 0.158i)19-s + (−0.0612 − 0.0163i)21-s + (−0.0730 + 0.0421i)23-s − 0.126·27-s − 1.34·29-s + (−0.00955 − 0.00551i)31-s + (−0.0467 − 0.0809i)33-s + (0.956 − 0.552i)37-s + (0.0217 − 0.0377i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7403180752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7403180752\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-12.7 - 47.3i)T \) |
| good | 3 | \( 1 + (0.285 - 0.493i)T + (-40.5 - 70.1i)T^{2} \) |
| 11 | \( 1 + (89.1 - 154. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + 116.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (182. - 316. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-98.9 + 57.1i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (38.6 - 22.3i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 1.12e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (9.18 + 5.30i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.30e3 + 756. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 515. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.07e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (806. + 1.39e3i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-2.25e3 - 1.30e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.96e3 - 2.28e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-4.87e3 + 2.81e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.19e3 + 1.84e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 813.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.33e3 + 4.04e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (633. + 1.09e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.06e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (8.33e3 - 4.81e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 - 9.56e3T + 8.85e7T^{2} \) |
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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
−10.29445902573582754779733859768, −9.590188207009754299032491518615, −8.610517272901099714659248178014, −7.69762668144533430589466798759, −7.04738463370071899675206532602, −5.68188720515875183329082771957, −5.00248381227199869026504839307, −4.07434051684421468527621524692, −2.37756288013286022292427659397, −1.90472026725764601981861096549,
0.18388482165160633967992554773, 1.08694910725535467962997288193, 2.69301978798389561616227020540, 3.74336828286547567574742326469, 4.73357577675393325508745800077, 5.77854721376342499608644251390, 6.86449050897954644426697403875, 7.53064017133178218603608917140, 8.447794115424424204221421362234, 9.537614938553878979242131989830
