| L(s) = 1 | + 0.151·2-s + 1.60·3-s − 1.97·4-s + 3.03·5-s + 0.242·6-s − 7-s − 0.601·8-s − 0.436·9-s + 0.459·10-s + 5.48·11-s − 3.16·12-s + 5.55·13-s − 0.151·14-s + 4.86·15-s + 3.86·16-s − 7.74·17-s − 0.0659·18-s − 0.179·19-s − 6.00·20-s − 1.60·21-s + 0.829·22-s − 3.54·23-s − 0.962·24-s + 4.22·25-s + 0.839·26-s − 5.50·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.106·2-s + 0.924·3-s − 0.988·4-s + 1.35·5-s + 0.0988·6-s − 0.377·7-s − 0.212·8-s − 0.145·9-s + 0.145·10-s + 1.65·11-s − 0.913·12-s + 1.54·13-s − 0.0403·14-s + 1.25·15-s + 0.965·16-s − 1.87·17-s − 0.0155·18-s − 0.0411·19-s − 1.34·20-s − 0.349·21-s + 0.176·22-s − 0.739·23-s − 0.196·24-s + 0.845·25-s + 0.164·26-s − 1.05·27-s + 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790347116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790347116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 0.151T + 2T^{2} \) |
| 3 | \( 1 - 1.60T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 + 0.179T + 19T^{2} \) |
| 23 | \( 1 + 3.54T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 + 6.35T + 37T^{2} \) |
| 41 | \( 1 + 7.26T + 41T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 7.70T + 53T^{2} \) |
| 59 | \( 1 + 0.943T + 59T^{2} \) |
| 61 | \( 1 + 2.50T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−11.84887803267300219271491968819, −10.55001674463372014117232760270, −9.493110908674885253787845254123, −8.911617258611004941398712241448, −8.503042182126556388053686153800, −6.54733575572110574589832395740, −5.95106802431246268724030401682, −4.37257370173083878237871579622, −3.34831371356513289967941662645, −1.73938576445286159955768372749,
1.73938576445286159955768372749, 3.34831371356513289967941662645, 4.37257370173083878237871579622, 5.95106802431246268724030401682, 6.54733575572110574589832395740, 8.503042182126556388053686153800, 8.911617258611004941398712241448, 9.493110908674885253787845254123, 10.55001674463372014117232760270, 11.84887803267300219271491968819
