| L(s) = 1 | + 0.0977·2-s − 2.00·3-s − 1.99·4-s + 4.15·5-s − 0.196·6-s − 1.90·7-s − 0.389·8-s + 1.03·9-s + 0.405·10-s − 11-s + 3.99·12-s + 3.91·13-s − 0.186·14-s − 8.34·15-s + 3.94·16-s + 0.101·18-s − 6.82·19-s − 8.26·20-s + 3.83·21-s − 0.0977·22-s − 2.14·23-s + 0.783·24-s + 12.2·25-s + 0.382·26-s + 3.94·27-s + 3.79·28-s − 4.94·29-s + ⋯ |
| L(s) = 1 | + 0.0690·2-s − 1.16·3-s − 0.995·4-s + 1.85·5-s − 0.0801·6-s − 0.720·7-s − 0.137·8-s + 0.345·9-s + 0.128·10-s − 0.301·11-s + 1.15·12-s + 1.08·13-s − 0.0497·14-s − 2.15·15-s + 0.985·16-s + 0.0238·18-s − 1.56·19-s − 1.84·20-s + 0.836·21-s − 0.0208·22-s − 0.448·23-s + 0.159·24-s + 2.45·25-s + 0.0749·26-s + 0.758·27-s + 0.717·28-s − 0.918·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067869475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067869475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 0.0977T + 2T^{2} \) |
| 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.550T + 67T^{2} \) |
| 71 | \( 1 + 3.42T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 0.655T + 79T^{2} \) |
| 83 | \( 1 + 0.269T + 83T^{2} \) |
| 89 | \( 1 - 0.177T + 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
| 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
−8.877571206331966830702614267828, −8.138354072845696909619994082999, −6.62687408015605604671024144007, −6.19216962260180725306966286766, −5.77542764186136412772999950004, −5.05902550863405213905072576072, −4.23135088252386892113932222895, −3.05541528714672882847164526341, −1.87398809693116269927674405904, −0.65929580347469130281818923367,
0.65929580347469130281818923367, 1.87398809693116269927674405904, 3.05541528714672882847164526341, 4.23135088252386892113932222895, 5.05902550863405213905072576072, 5.77542764186136412772999950004, 6.19216962260180725306966286766, 6.62687408015605604671024144007, 8.138354072845696909619994082999, 8.877571206331966830702614267828
