L(s) = 1 | − 0.929i·5-s − 9.68·11-s + 15.9i·13-s − 10.5i·17-s − 7.22i·19-s + 11.3·23-s + 24.1·25-s − 46.3·29-s − 0.483i·31-s + 2.48·37-s − 55.8i·41-s + 60.6·43-s + 36.5i·47-s + 28.5·53-s + 9.00i·55-s + ⋯ |
L(s) = 1 | − 0.185i·5-s − 0.880·11-s + 1.22i·13-s − 0.620i·17-s − 0.380i·19-s + 0.491·23-s + 0.965·25-s − 1.59·29-s − 0.0155i·31-s + 0.0670·37-s − 1.36i·41-s + 1.41·43-s + 0.778i·47-s + 0.538·53-s + 0.163i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.336127784\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.336127784\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.929iT - 25T^{2} \) |
| 11 | \( 1 + 9.68T + 121T^{2} \) |
| 13 | \( 1 - 15.9iT - 169T^{2} \) |
| 17 | \( 1 + 10.5iT - 289T^{2} \) |
| 19 | \( 1 + 7.22iT - 361T^{2} \) |
| 23 | \( 1 - 11.3T + 529T^{2} \) |
| 29 | \( 1 + 46.3T + 841T^{2} \) |
| 31 | \( 1 + 0.483iT - 961T^{2} \) |
| 37 | \( 1 - 2.48T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 94.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 127.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 46.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 18.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 59.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 71.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 102. iT - 9.40e3T^{2} \) |
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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
−9.235008391869103656432627458250, −8.126402057007061495602555256448, −7.30359450351799445828836770585, −6.68517764106042495566330851709, −5.56224994508310896569421587180, −4.88447047680753351896067733690, −3.96887282619269989060046351399, −2.82954233810510214638255459755, −1.85371833843514582864631016371, −0.39711982019141048411410828179,
1.02894165830333836472010570570, 2.45198050233381844880525203811, 3.25632815236009787288309875633, 4.30897859277836407013585214565, 5.42222754504183106919850466086, 5.86433142084335624790037960303, 7.06363453620470817460154982219, 7.71142752037606017569936677713, 8.432090796241675810833749519832, 9.252740879820639605921599868593