| L(s) = 1 | − 0.319·2-s − 2.37·3-s − 1.89·4-s − 0.869·5-s + 0.757·6-s + 1.24·8-s + 2.61·9-s + 0.277·10-s − 5.82·11-s + 4.49·12-s − 6.07·13-s + 2.06·15-s + 3.39·16-s + 3.26·17-s − 0.836·18-s + 2.35·19-s + 1.65·20-s + 1.86·22-s − 2.60·23-s − 2.95·24-s − 4.24·25-s + 1.94·26-s + 0.905·27-s + 1.00·29-s − 0.658·30-s − 3.63·31-s − 3.57·32-s + ⋯ |
| L(s) = 1 | − 0.225·2-s − 1.36·3-s − 0.948·4-s − 0.388·5-s + 0.309·6-s + 0.440·8-s + 0.872·9-s + 0.0878·10-s − 1.75·11-s + 1.29·12-s − 1.68·13-s + 0.532·15-s + 0.849·16-s + 0.791·17-s − 0.197·18-s + 0.539·19-s + 0.369·20-s + 0.397·22-s − 0.543·23-s − 0.602·24-s − 0.848·25-s + 0.380·26-s + 0.174·27-s + 0.186·29-s − 0.120·30-s − 0.653·31-s − 0.632·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.319T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 0.869T + 5T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 - 6.18T + 47T^{2} \) |
| 53 | \( 1 + 4.16T + 53T^{2} \) |
| 59 | \( 1 + 0.882T + 59T^{2} \) |
| 61 | \( 1 - 0.377T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.62T + 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
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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
−7.64359042090140103168677897016, −7.25897876869119538554025996050, −5.95907047577842497507364484144, −5.45296137606770539367113895660, −4.96700872823276141679428163200, −4.37948622038046869704600590727, −3.29531343455096997009525248670, −2.23677921845704650890961855113, −0.70661141588832120694866512210, 0,
0.70661141588832120694866512210, 2.23677921845704650890961855113, 3.29531343455096997009525248670, 4.37948622038046869704600590727, 4.96700872823276141679428163200, 5.45296137606770539367113895660, 5.95907047577842497507364484144, 7.25897876869119538554025996050, 7.64359042090140103168677897016
