| L(s) = 1 | − 0.273·2-s + 2.65·3-s − 1.92·4-s − 0.726·6-s − 0.273·7-s + 1.07·8-s + 4.02·9-s + 1.37·11-s − 5.10·12-s − 2.75·13-s + 0.0750·14-s + 3.55·16-s + 2·17-s − 1.10·18-s + 4.20·19-s − 0.726·21-s − 0.377·22-s + 8.05·23-s + 2.84·24-s + 0.754·26-s + 2.72·27-s + 0.527·28-s − 4.20·29-s + 10.8·31-s − 3.12·32-s + 3.65·33-s − 0.547·34-s + ⋯ |
| L(s) = 1 | − 0.193·2-s + 1.53·3-s − 0.962·4-s − 0.296·6-s − 0.103·7-s + 0.380·8-s + 1.34·9-s + 0.415·11-s − 1.47·12-s − 0.763·13-s + 0.0200·14-s + 0.888·16-s + 0.485·17-s − 0.260·18-s + 0.965·19-s − 0.158·21-s − 0.0804·22-s + 1.67·23-s + 0.581·24-s + 0.147·26-s + 0.524·27-s + 0.0996·28-s − 0.781·29-s + 1.95·31-s − 0.552·32-s + 0.635·33-s − 0.0939·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091666120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091666120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.273T + 2T^{2} \) |
| 3 | \( 1 - 2.65T + 3T^{2} \) |
| 7 | \( 1 + 0.273T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 8.05T + 23T^{2} \) |
| 29 | \( 1 + 4.20T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 5.92T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 + 0.0467T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 0.281T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 0.341T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.556038382374739686740827342535, −9.173362288939965375261867306942, −8.293030556761268345251390811107, −7.72606349863759642970470674990, −6.85572182670984527775954655477, −5.36568447725688986152841543350, −4.48402600365511601052728325760, −3.48385961322679025797421649626, −2.71985571825377468111213432248, −1.17559450468561768005822487905,
1.17559450468561768005822487905, 2.71985571825377468111213432248, 3.48385961322679025797421649626, 4.48402600365511601052728325760, 5.36568447725688986152841543350, 6.85572182670984527775954655477, 7.72606349863759642970470674990, 8.293030556761268345251390811107, 9.173362288939965375261867306942, 9.556038382374739686740827342535
