| L(s) = 1 | + 5.14i·5-s − 12.6i·7-s + 1.46·11-s − 5.68i·13-s − 14.2·17-s − 26.6·19-s + 36.7i·23-s − 1.42·25-s − 19.4i·29-s + 16.1i·31-s + 65.0·35-s + 37.2i·37-s − 58.8·41-s − 61.2·43-s − 61.6i·47-s + ⋯ |
| L(s) = 1 | + 1.02i·5-s − 1.80i·7-s + 0.132·11-s − 0.437i·13-s − 0.839·17-s − 1.40·19-s + 1.59i·23-s − 0.0570·25-s − 0.672i·29-s + 0.522i·31-s + 1.85·35-s + 1.00i·37-s − 1.43·41-s − 1.42·43-s − 1.31i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2479110450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2479110450\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.14iT - 25T^{2} \) |
| 7 | \( 1 + 12.6iT - 49T^{2} \) |
| 11 | \( 1 - 1.46T + 121T^{2} \) |
| 13 | \( 1 + 5.68iT - 169T^{2} \) |
| 17 | \( 1 + 14.2T + 289T^{2} \) |
| 19 | \( 1 + 26.6T + 361T^{2} \) |
| 23 | \( 1 - 36.7iT - 529T^{2} \) |
| 29 | \( 1 + 19.4iT - 841T^{2} \) |
| 31 | \( 1 - 16.1iT - 961T^{2} \) |
| 37 | \( 1 - 37.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 61.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 61.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 42.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 2.74T + 3.48e3T^{2} \) |
| 61 | \( 1 + 71.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 10.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 82.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 68.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 65.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 5.31T + 9.40e3T^{2} \) |
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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
−10.03074139541232801151597351250, −8.612965162833112630463970142846, −7.73097354510665164791195541570, −6.84327098063404070789756202010, −6.52157766241998041787202727045, −4.99566456244117619070886864263, −3.94388028749666936104942928750, −3.21817837974412708011673584145, −1.70521168147824514033297483037, −0.07653202139162420182552411225,
1.78089420822010844482236829134, 2.68409021425301476029773411128, 4.32103504047407591473239583038, 5.00090962235650544879922785508, 6.00403843200712974758449766259, 6.70613736053342787987128954396, 8.227987671733126380068820492598, 8.876646215898123863663537726105, 9.034864079695583869727378902172, 10.32107128304024348287883130670
