L(s) = 1 | − 1.54·2-s + 0.398·4-s − 0.334·5-s − 4.24·7-s + 2.47·8-s + 0.518·10-s − 0.915·11-s + 4.81·13-s + 6.57·14-s − 4.63·16-s + 5.38·17-s − 4.34·19-s − 0.133·20-s + 1.41·22-s − 8.10·23-s − 4.88·25-s − 7.45·26-s − 1.69·28-s + 6.45·29-s + 10.7·31-s + 2.22·32-s − 8.34·34-s + 1.42·35-s + 5.16·37-s + 6.73·38-s − 0.830·40-s + 0.612·41-s + ⋯ |
L(s) = 1 | − 1.09·2-s + 0.199·4-s − 0.149·5-s − 1.60·7-s + 0.876·8-s + 0.164·10-s − 0.276·11-s + 1.33·13-s + 1.75·14-s − 1.15·16-s + 1.30·17-s − 0.997·19-s − 0.0298·20-s + 0.302·22-s − 1.69·23-s − 0.977·25-s − 1.46·26-s − 0.319·28-s + 1.19·29-s + 1.93·31-s + 0.393·32-s − 1.43·34-s + 0.240·35-s + 0.849·37-s + 1.09·38-s − 0.131·40-s + 0.0956·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 5 | \( 1 + 0.334T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.915T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 8.10T + 23T^{2} \) |
| 29 | \( 1 - 6.45T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 - 0.612T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 + 0.00846T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 - 0.520T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−8.518730259851255371145912088817, −8.238677721515726289823926023467, −7.34160535147939316760343433975, −6.24109767550795764150821567092, −6.00328357204186478554765697071, −4.42040481849958117264709650213, −3.69135537695197726923674999683, −2.63502179988512222898405046868, −1.18747882564595536654529453565, 0,
1.18747882564595536654529453565, 2.63502179988512222898405046868, 3.69135537695197726923674999683, 4.42040481849958117264709650213, 6.00328357204186478554765697071, 6.24109767550795764150821567092, 7.34160535147939316760343433975, 8.238677721515726289823926023467, 8.518730259851255371145912088817