| L(s) = 1 | − 0.711·2-s − 2.33·3-s − 1.49·4-s − 5-s + 1.66·6-s − 3.28·7-s + 2.48·8-s + 2.45·9-s + 0.711·10-s − 11-s + 3.48·12-s − 1.30·13-s + 2.34·14-s + 2.33·15-s + 1.21·16-s − 4.80·17-s − 1.74·18-s − 6.34·19-s + 1.49·20-s + 7.68·21-s + 0.711·22-s + 0.965·23-s − 5.80·24-s + 25-s + 0.929·26-s + 1.26·27-s + 4.91·28-s + ⋯ |
| L(s) = 1 | − 0.503·2-s − 1.34·3-s − 0.746·4-s − 0.447·5-s + 0.678·6-s − 1.24·7-s + 0.879·8-s + 0.819·9-s + 0.225·10-s − 0.301·11-s + 1.00·12-s − 0.362·13-s + 0.625·14-s + 0.603·15-s + 0.304·16-s − 1.16·17-s − 0.412·18-s − 1.45·19-s + 0.333·20-s + 1.67·21-s + 0.151·22-s + 0.201·23-s − 1.18·24-s + 0.200·25-s + 0.182·26-s + 0.243·27-s + 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.711T + 2T^{2} \) |
| 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 0.965T + 23T^{2} \) |
| 29 | \( 1 - 0.174T + 29T^{2} \) |
| 31 | \( 1 - 6.72T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 - 1.00T + 41T^{2} \) |
| 43 | \( 1 - 0.999T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 - 5.15T + 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 7.19T + 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−8.189232441275345808842416706099, −7.25794093892339765788896720737, −6.43624533754723660844898622650, −6.11235849028933576139475589999, −4.89976435077773775083880512055, −4.55168756337448443077664326320, −3.60110481009371822121575818848, −2.37185923307834836366232258021, −0.71554540224512612497411001076, 0,
0.71554540224512612497411001076, 2.37185923307834836366232258021, 3.60110481009371822121575818848, 4.55168756337448443077664326320, 4.89976435077773775083880512055, 6.11235849028933576139475589999, 6.43624533754723660844898622650, 7.25794093892339765788896720737, 8.189232441275345808842416706099
