| L(s) = 1 | + 7.98·5-s + 2.13i·7-s + 8i·11-s − 11.6·13-s − 11.8·17-s − 14.9i·19-s + 4.27i·23-s + 38.7·25-s + 0.573·29-s + 57.4i·31-s + 17.0i·35-s + 27.6·37-s − 31.5·41-s + 28.7i·43-s + 59.5i·47-s + ⋯ |
| L(s) = 1 | + 1.59·5-s + 0.305i·7-s + 0.727i·11-s − 0.898·13-s − 0.697·17-s − 0.785i·19-s + 0.185i·23-s + 1.54·25-s + 0.0197·29-s + 1.85i·31-s + 0.487i·35-s + 0.747·37-s − 0.769·41-s + 0.669i·43-s + 1.26i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.189216167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189216167\) |
| \(L(2)\) |
|
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 | \( 1 - 7.98T + 25T^{2} \) |
| 7 | \( 1 - 2.13iT - 49T^{2} \) |
| 11 | \( 1 - 8iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 19 | \( 1 + 14.9iT - 361T^{2} \) |
| 23 | \( 1 - 4.27iT - 529T^{2} \) |
| 29 | \( 1 - 0.573T + 841T^{2} \) |
| 31 | \( 1 - 57.4iT - 961T^{2} \) |
| 37 | \( 1 - 27.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 31.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 52.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 84.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.42T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 67.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 97.1T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.177591115888689537852351459361, −8.458671438096312722823567400271, −7.22380515923834105850212123033, −6.73019116227374896868951342402, −5.84951463709926552568832539275, −5.09731708929070027067071342475, −4.46613152252983114967426138580, −2.87560054523577491566829340616, −2.27705629567074062181995745072, −1.31322033863490682622071598696,
0.49196295727260364503166107588, 1.86626294657060002656161921648, 2.49508125091774262193881740423, 3.69680511748287327646075795797, 4.77849554239421770105960048405, 5.61089929585375249926593823660, 6.18623260350312361517106857498, 6.94553704205014944643723037507, 7.906129878919669346674902978647, 8.747148366690613878107659435835
