| L(s) = 1 | − 1.81·2-s + 1.30·4-s + 1.39·5-s − 7-s + 1.26·8-s − 2.54·10-s − 0.0844·11-s − 3.52·13-s + 1.81·14-s − 4.90·16-s − 4.93·17-s − 3.17·19-s + 1.82·20-s + 0.153·22-s + 3.91·23-s − 3.04·25-s + 6.40·26-s − 1.30·28-s + 7.80·29-s − 6.27·31-s + 6.39·32-s + 8.97·34-s − 1.39·35-s + 3.42·37-s + 5.77·38-s + 1.76·40-s + 9.78·41-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.652·4-s + 0.625·5-s − 0.377·7-s + 0.446·8-s − 0.803·10-s − 0.0254·11-s − 0.977·13-s + 0.485·14-s − 1.22·16-s − 1.19·17-s − 0.729·19-s + 0.407·20-s + 0.0327·22-s + 0.817·23-s − 0.609·25-s + 1.25·26-s − 0.246·28-s + 1.44·29-s − 1.12·31-s + 1.13·32-s + 1.53·34-s − 0.236·35-s + 0.563·37-s + 0.937·38-s + 0.279·40-s + 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 11 | \( 1 + 0.0844T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 - 0.234T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 - 5.09T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 9.13T + 97T^{2} \) |
| 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
−7.63876536402003024065639073517, −6.87078864923516897171173316390, −6.43829233185253621239102634991, −5.48428988221368948313868087815, −4.65752217533074402286286397121, −3.99400257189717213252332564911, −2.54352294292590239521241751780, −2.23502530525214066427846808923, −1.03356611101232736131599673483, 0,
1.03356611101232736131599673483, 2.23502530525214066427846808923, 2.54352294292590239521241751780, 3.99400257189717213252332564911, 4.65752217533074402286286397121, 5.48428988221368948313868087815, 6.43829233185253621239102634991, 6.87078864923516897171173316390, 7.63876536402003024065639073517
