| L(s) = 1 | − 1.73·2-s + 2.13·3-s + 1.00·4-s − 4.15·5-s − 3.69·6-s − 1.81·7-s + 1.72·8-s + 1.53·9-s + 7.20·10-s + 11-s + 2.14·12-s + 3.34·13-s + 3.15·14-s − 8.86·15-s − 4.99·16-s − 17-s − 2.66·18-s − 0.395·19-s − 4.18·20-s − 3.87·21-s − 1.73·22-s + 3.74·23-s + 3.67·24-s + 12.2·25-s − 5.79·26-s − 3.11·27-s − 1.82·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.22·3-s + 0.502·4-s − 1.86·5-s − 1.50·6-s − 0.687·7-s + 0.609·8-s + 0.512·9-s + 2.27·10-s + 0.301·11-s + 0.618·12-s + 0.926·13-s + 0.842·14-s − 2.28·15-s − 1.24·16-s − 0.242·17-s − 0.628·18-s − 0.0907·19-s − 0.934·20-s − 0.845·21-s − 0.369·22-s + 0.781·23-s + 0.749·24-s + 2.45·25-s − 1.13·26-s − 0.599·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7779059054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7779059054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 19 | \( 1 + 0.395T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 47 | \( 1 - 0.482T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 0.787T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.909591612336868823494148818478, −7.59233835005443692948972418919, −6.89602411933576526172194140015, −6.09949821774437621481818020821, −4.51740309104439124248278745170, −4.23072978364773121137786186559, −3.23562374097783842764154976589, −2.90781718626815409501303759300, −1.50001148474749690564601568287, −0.51990435895031157928822570556,
0.51990435895031157928822570556, 1.50001148474749690564601568287, 2.90781718626815409501303759300, 3.23562374097783842764154976589, 4.23072978364773121137786186559, 4.51740309104439124248278745170, 6.09949821774437621481818020821, 6.89602411933576526172194140015, 7.59233835005443692948972418919, 7.909591612336868823494148818478
