L(s) = 1 | + 0.299·2-s − 1.73·3-s − 3.91·4-s + 2.23i·5-s − 0.518·6-s + 2.64i·7-s − 2.36·8-s + 2.99·9-s + 0.670i·10-s + 17.7i·11-s + 6.77·12-s − 2.29·13-s + 0.792i·14-s − 3.87i·15-s + 14.9·16-s − 25.6i·17-s + ⋯ |
L(s) = 1 | + 0.149·2-s − 0.577·3-s − 0.977·4-s + 0.447i·5-s − 0.0864·6-s + 0.377i·7-s − 0.296·8-s + 0.333·9-s + 0.0670i·10-s + 1.61i·11-s + 0.564·12-s − 0.176·13-s + 0.0565i·14-s − 0.258i·15-s + 0.933·16-s − 1.50i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2857434948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2857434948\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (20.7 + 9.97i)T \) |
good | 2 | \( 1 - 0.299T + 4T^{2} \) |
| 5 | \( 1 - 2.23iT - 25T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 2.29T + 169T^{2} \) |
| 17 | \( 1 + 25.6iT - 289T^{2} \) |
| 19 | \( 1 + 5.56iT - 361T^{2} \) |
| 29 | \( 1 + 32.8T + 841T^{2} \) |
| 31 | \( 1 + 53.8T + 961T^{2} \) |
| 37 | \( 1 + 11.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 72.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 72.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 98.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 7.20iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 43.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 24.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 21.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 59.2iT - 9.40e3T^{2} \) |
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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
−10.37769458789507724431564432025, −9.604404395297016731349337277833, −8.955000383046955233745648436787, −7.53561953011952541733896272513, −6.88308145424456377677741670912, −5.48596306795079780801153502302, −4.86995260647218398551870248810, −3.78181449681737360709860421763, −2.19061273593235254791253251136, −0.13621000466488362537325319568,
1.23616663722823140406055758689, 3.51796047153133151249724724369, 4.27577870487787338543125898298, 5.57959964081272471669911601008, 5.97306669359694930064872416590, 7.54733579359503752338858888115, 8.497123502697520263285241429095, 9.154911956012431042103016247701, 10.30433808147332937484293592006, 10.95455397837877923837338191320