| L(s) = 1 | − 0.456·2-s − 0.835·3-s − 1.79·4-s + 0.221·5-s + 0.381·6-s + 4.69·7-s + 1.73·8-s − 2.30·9-s − 0.101·10-s − 11-s + 1.49·12-s + 5.89·13-s − 2.14·14-s − 0.185·15-s + 2.79·16-s + 7.06·17-s + 1.05·18-s + 19-s − 0.397·20-s − 3.92·21-s + 0.456·22-s + 1.06·23-s − 1.44·24-s − 4.95·25-s − 2.68·26-s + 4.42·27-s − 8.41·28-s + ⋯ |
| L(s) = 1 | − 0.322·2-s − 0.482·3-s − 0.895·4-s + 0.0992·5-s + 0.155·6-s + 1.77·7-s + 0.612·8-s − 0.767·9-s − 0.0320·10-s − 0.301·11-s + 0.431·12-s + 1.63·13-s − 0.573·14-s − 0.0478·15-s + 0.698·16-s + 1.71·17-s + 0.247·18-s + 0.229·19-s − 0.0889·20-s − 0.856·21-s + 0.0973·22-s + 0.221·23-s − 0.295·24-s − 0.990·25-s − 0.527·26-s + 0.852·27-s − 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8845142238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8845142238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.456T + 2T^{2} \) |
| 3 | \( 1 + 0.835T + 3T^{2} \) |
| 5 | \( 1 - 0.221T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 + 7.62T + 29T^{2} \) |
| 31 | \( 1 - 0.901T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 - 0.788T + 41T^{2} \) |
| 43 | \( 1 - 0.714T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.14T + 97T^{2} \) |
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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
−12.19523134593506336933721136825, −11.24315590331099936105207434402, −10.63960423743970911078648786137, −9.319308212683367048424991081905, −8.260618365977527678505227769444, −7.80739518562844672234277678145, −5.74970565770299766656927693801, −5.17842296779522439176261900762, −3.76566986960473015694358687349, −1.32305777517800720586171586944,
1.32305777517800720586171586944, 3.76566986960473015694358687349, 5.17842296779522439176261900762, 5.74970565770299766656927693801, 7.80739518562844672234277678145, 8.260618365977527678505227769444, 9.319308212683367048424991081905, 10.63960423743970911078648786137, 11.24315590331099936105207434402, 12.19523134593506336933721136825
