| L(s) = 1 | + 4.47i·5-s − 9.32·7-s + 19.0i·11-s − 23.9·13-s − 30.3i·17-s + 21.6·19-s − 13.4i·23-s + 4.99·25-s − 42.8i·29-s + 19.2·31-s − 41.7i·35-s + 25.9·37-s + 10.7i·41-s − 32.5·43-s − 28.9i·47-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s − 1.33·7-s + 1.73i·11-s − 1.84·13-s − 1.78i·17-s + 1.13·19-s − 0.583i·23-s + 0.199·25-s − 1.47i·29-s + 0.622·31-s − 1.19i·35-s + 0.701·37-s + 0.262i·41-s − 0.758·43-s − 0.615i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5552317120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5552317120\) |
| \(L(2)\) |
|
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 | \( 1 - 4.47iT - 25T^{2} \) |
| 7 | \( 1 + 9.32T + 49T^{2} \) |
| 11 | \( 1 - 19.0iT - 121T^{2} \) |
| 13 | \( 1 + 23.9T + 169T^{2} \) |
| 17 | \( 1 + 30.3iT - 289T^{2} \) |
| 19 | \( 1 - 21.6T + 361T^{2} \) |
| 23 | \( 1 + 13.4iT - 529T^{2} \) |
| 29 | \( 1 + 42.8iT - 841T^{2} \) |
| 31 | \( 1 - 19.2T + 961T^{2} \) |
| 37 | \( 1 - 25.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 32.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 28.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 58.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.02T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 2.90iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 32.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 41.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 116. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 5.05T + 9.40e3T^{2} \) |
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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
−9.867190935112600139484273051742, −9.392235421478455349923191420172, −7.69133330975330313858088481815, −7.05734641868560598149574728678, −6.67398193956104723782106450900, −5.24716315765645375948911377310, −4.40495153437006374972599248386, −2.90386255949169844673606575618, −2.48352160256755629077886219502, −0.20279621277503081602205843170,
1.10453917596096461645815789925, 2.87394610637789981164479373316, 3.65828549424655762143028133808, 4.98960073697254520735105318037, 5.77968109785528714197548504778, 6.63523238446375620733744329127, 7.72392246127413743560448644503, 8.599742722051412229665647984062, 9.340708313690051217989239602456, 10.02171236087980636811908860898
