| L(s) = 1 | − 1.70·2-s + 3-s + 0.903·4-s − 1.70·6-s − 0.338·7-s + 1.86·8-s + 9-s + 1.26·11-s + 0.903·12-s + 6.57·13-s + 0.576·14-s − 4.99·16-s + 3.65·17-s − 1.70·18-s + 19-s − 0.338·21-s − 2.14·22-s − 5.14·23-s + 1.86·24-s − 11.1·26-s + 27-s − 0.305·28-s − 4.73·29-s − 3.25·31-s + 4.76·32-s + 1.26·33-s − 6.23·34-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.577·3-s + 0.451·4-s − 0.695·6-s − 0.127·7-s + 0.660·8-s + 0.333·9-s + 0.380·11-s + 0.260·12-s + 1.82·13-s + 0.154·14-s − 1.24·16-s + 0.887·17-s − 0.401·18-s + 0.229·19-s − 0.0738·21-s − 0.458·22-s − 1.07·23-s + 0.381·24-s − 2.19·26-s + 0.192·27-s − 0.0577·28-s − 0.879·29-s − 0.583·31-s + 0.842·32-s + 0.219·33-s − 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174979197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174979197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 7 | \( 1 + 0.338T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 6.57T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 0.160T + 67T^{2} \) |
| 71 | \( 1 - 0.160T + 71T^{2} \) |
| 73 | \( 1 + 5.07T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.404497255010487217130927612185, −8.796425043342278678671839852022, −8.085967473918548208335126339971, −7.52606432920702913142274615568, −6.48657042688413364641759715778, −5.58709418214912218204102566134, −4.15366271845764524040492103983, −3.47614486751650999411694429390, −1.96669181962175537554896908957, −0.974895974358141710390128569315,
0.974895974358141710390128569315, 1.96669181962175537554896908957, 3.47614486751650999411694429390, 4.15366271845764524040492103983, 5.58709418214912218204102566134, 6.48657042688413364641759715778, 7.52606432920702913142274615568, 8.085967473918548208335126339971, 8.796425043342278678671839852022, 9.404497255010487217130927612185
