| L(s) = 1 | − 2-s − 3-s + 4-s − 0.732·5-s + 6-s + 0.267·7-s − 8-s − 2·9-s + 0.732·10-s − 12-s − 3.46·13-s − 0.267·14-s + 0.732·15-s + 16-s − 7.46·17-s + 2·18-s + 19-s − 0.732·20-s − 0.267·21-s − 23-s + 24-s − 4.46·25-s + 3.46·26-s + 5·27-s + 0.267·28-s + 1.73·29-s − 0.732·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.327·5-s + 0.408·6-s + 0.101·7-s − 0.353·8-s − 0.666·9-s + 0.231·10-s − 0.288·12-s − 0.960·13-s − 0.0716·14-s + 0.189·15-s + 0.250·16-s − 1.81·17-s + 0.471·18-s + 0.229·19-s − 0.163·20-s − 0.0584·21-s − 0.208·23-s + 0.204·24-s − 0.892·25-s + 0.679·26-s + 0.962·27-s + 0.0506·28-s + 0.321·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3362056027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3362056027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 - 0.267T + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 1.26T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
| 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
−8.321097595735992382980196174602, −7.68840621609576732569336126168, −6.82272702198640737011776094892, −6.37262799311460060404166761712, −5.38266803331276432754755090571, −4.76852537712280093870554148054, −3.74924779892313023992806743840, −2.66527301920239186644644347413, −1.87687307597278187780052376503, −0.34922137815240737037392207977,
0.34922137815240737037392207977, 1.87687307597278187780052376503, 2.66527301920239186644644347413, 3.74924779892313023992806743840, 4.76852537712280093870554148054, 5.38266803331276432754755090571, 6.37262799311460060404166761712, 6.82272702198640737011776094892, 7.68840621609576732569336126168, 8.321097595735992382980196174602
