L(s) = 1 | + 3.06i·2-s + (0.126 − 2.99i)3-s − 5.36·4-s − 6.63i·5-s + (9.17 + 0.386i)6-s + 3.06·7-s − 4.17i·8-s + (−8.96 − 0.757i)9-s + 20.3·10-s + (−0.678 + 16.0i)12-s − 16.2·13-s + 9.37i·14-s + (−19.9 − 0.839i)15-s − 8.67·16-s − 0.805i·17-s + (2.31 − 27.4i)18-s + ⋯ |
L(s) = 1 | + 1.53i·2-s + (0.0421 − 0.999i)3-s − 1.34·4-s − 1.32i·5-s + (1.52 + 0.0644i)6-s + 0.437·7-s − 0.522i·8-s + (−0.996 − 0.0842i)9-s + 2.03·10-s + (−0.0565 + 1.34i)12-s − 1.24·13-s + 0.669i·14-s + (−1.32 − 0.0559i)15-s − 0.542·16-s − 0.0473i·17-s + (0.128 − 1.52i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0421 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0421 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.513843 - 0.535975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.513843 - 0.535975i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.126 + 2.99i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 3.06iT - 4T^{2} \) |
| 5 | \( 1 + 6.63iT - 25T^{2} \) |
| 7 | \( 1 - 3.06T + 49T^{2} \) |
| 13 | \( 1 + 16.2T + 169T^{2} \) |
| 17 | \( 1 + 0.805iT - 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 + 27.3iT - 529T^{2} \) |
| 29 | \( 1 + 3.78iT - 841T^{2} \) |
| 31 | \( 1 + 20.7T + 961T^{2} \) |
| 37 | \( 1 - 38.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 43.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 72.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 2.56iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 98.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 31.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 63.3T + 9.40e3T^{2} \) |
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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
−11.17034999957413913109910225287, −9.524733328495492631855878295141, −8.535955736868345023193366569947, −8.149442674787557345054508124193, −7.18728293537845106467567709172, −6.30022073411976282527931819683, −5.24226296614525266019162054897, −4.55985029165700557589334604139, −2.12130614992750371230511869172, −0.31181798973931263834061539966,
2.17579587241332748985966451945, 3.07148516958043159948512179130, 4.03449724410623111097100377587, 5.11789590374221328042336086085, 6.62999597055507027584441975962, 7.923870465532814922406385419324, 9.285075701495675834283583695190, 9.920186038385299430343734148986, 10.67198172244387524540220791254, 11.21074630723069739495865173119