L(s) = 1 | + 2.82i·2-s − 5.19·3-s − 8.00·4-s + 6.35·5-s − 14.6i·6-s − 12.8·7-s − 22.6i·8-s + 27·9-s + 17.9i·10-s − 20.6i·11-s + 41.5·12-s − 24.6i·13-s − 36.4i·14-s − 33.0·15-s + 64.0·16-s + 132.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.254·5-s − 0.408i·6-s − 0.262·7-s − 0.353i·8-s + 0.333·9-s + 0.179i·10-s − 0.170i·11-s + 0.288·12-s − 0.145i·13-s − 0.185i·14-s − 0.146·15-s + 0.250·16-s + 0.458·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8261946323\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8261946323\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (1.05e3 - 3.31e3i)T \) |
good | 5 | \( 1 - 6.35T + 625T^{2} \) |
| 7 | \( 1 + 12.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 20.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 24.6iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 132.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 240.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 545. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 873.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 984. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.55e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 42.3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.37e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 691. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.28e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.15e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.35e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.96e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 7.26e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 9.99e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 390. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 978. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.10e4iT - 8.85e7T^{2} \) |
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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
−11.35908656186300151954527018966, −10.05060829589212578946143363887, −9.578591883527306447111712118554, −8.260017677295267605516203753735, −7.40464466982297894736664906576, −6.31133834853306821208453325111, −5.61960018204258332474048011179, −4.55598344311648645883352791600, −3.18491356549770966393592694786, −1.24109034150744936248754313013,
0.28682751491075065157201938502, 1.66220759751527897918797442839, 3.06026078454493572254023475156, 4.33900861986274097690523955665, 5.39432867605961952831962707295, 6.41608039830125113375160597179, 7.57783193509807251857342172592, 8.797455800784404651126334618568, 9.772005270383154479386354169753, 10.43500274305867337512406216052