| L(s) = 1 | − 1.71i·2-s − 1.73·3-s + 1.06·4-s + 2.96i·6-s + (−6.15 − 3.33i)7-s − 8.67i·8-s + 2.99·9-s + 17.0·11-s − 1.85·12-s + 16.3·13-s + (−5.70 + 10.5i)14-s − 10.5·16-s − 13.4·17-s − 5.13i·18-s − 13.7i·19-s + ⋯ |
| L(s) = 1 | − 0.856i·2-s − 0.577·3-s + 0.267·4-s + 0.494i·6-s + (−0.879 − 0.476i)7-s − 1.08i·8-s + 0.333·9-s + 1.54·11-s − 0.154·12-s + 1.25·13-s + (−0.407 + 0.752i)14-s − 0.661·16-s − 0.789·17-s − 0.285i·18-s − 0.723i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.459175580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459175580\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.15 + 3.33i)T \) |
| good | 2 | \( 1 + 1.71iT - 4T^{2} \) |
| 11 | \( 1 - 17.0T + 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 + 13.4T + 289T^{2} \) |
| 19 | \( 1 + 13.7iT - 361T^{2} \) |
| 23 | \( 1 - 16.6iT - 529T^{2} \) |
| 29 | \( 1 + 32.1T + 841T^{2} \) |
| 31 | \( 1 + 6.74iT - 961T^{2} \) |
| 37 | \( 1 + 69.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 22.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 81.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 72.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 75.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 128. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 159.T + 9.40e3T^{2} \) |
| 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
−10.58383682293336446595788813712, −9.519521753262647556362241277554, −8.971098262422051883539866920674, −7.20591551578510933705915983588, −6.64925991679130547773024832802, −5.81968155192783167303485240659, −4.02519934383836347609798559107, −3.57007465478490768041762519429, −1.88498185331064984327030064814, −0.63543659123142998489759966516,
1.56850144700465152792798897939, 3.27889568741150490768303335506, 4.54629611389963711240027280335, 6.07559670198715573055153113706, 6.20561832383158975471394093898, 7.02657767771254624160948010797, 8.329838419829836528288970066089, 9.043441246435618275034299200139, 10.08255395693269118769199663193, 11.25224955388925509351167203070
