| L(s) = 1 | − 0.642·2-s − 3-s − 1.58·4-s + 1.17·5-s + 0.642·6-s − 0.0115·7-s + 2.30·8-s + 9-s − 0.755·10-s + 4.09·11-s + 1.58·12-s + 4.84·13-s + 0.00740·14-s − 1.17·15-s + 1.69·16-s + 17-s − 0.642·18-s + 4.74·19-s − 1.86·20-s + 0.0115·21-s − 2.63·22-s − 5.76·23-s − 2.30·24-s − 3.61·25-s − 3.11·26-s − 27-s + 0.0182·28-s + ⋯ |
| L(s) = 1 | − 0.454·2-s − 0.577·3-s − 0.793·4-s + 0.525·5-s + 0.262·6-s − 0.00435·7-s + 0.815·8-s + 0.333·9-s − 0.238·10-s + 1.23·11-s + 0.458·12-s + 1.34·13-s + 0.00197·14-s − 0.303·15-s + 0.422·16-s + 0.242·17-s − 0.151·18-s + 1.08·19-s − 0.416·20-s + 0.00251·21-s − 0.561·22-s − 1.20·23-s − 0.470·24-s − 0.723·25-s − 0.610·26-s − 0.192·27-s + 0.00345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 0.642T + 2T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 + 0.0115T + 7T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 6.25T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + 6.35T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 5.91T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.56976070757072630514807558526, −6.73838558894131733430202450411, −6.00246606029080105093701162945, −5.55909565189324258795771881778, −4.70019508712562480385936179239, −3.86385627073529161454435297207, −3.40457153241713150027148950706, −1.65382792711741180683003811528, −1.32717137823736535580855971079, 0,
1.32717137823736535580855971079, 1.65382792711741180683003811528, 3.40457153241713150027148950706, 3.86385627073529161454435297207, 4.70019508712562480385936179239, 5.55909565189324258795771881778, 6.00246606029080105093701162945, 6.73838558894131733430202450411, 7.56976070757072630514807558526
