| L(s) = 1 | − 2.34·2-s + 2.19·3-s + 3.48·4-s − 5.14·6-s + 1.27·7-s − 3.49·8-s + 1.81·9-s − 2.28·11-s + 7.66·12-s + 5.85·13-s − 2.98·14-s + 1.19·16-s − 1.38·17-s − 4.26·18-s − 0.503·19-s + 2.79·21-s + 5.35·22-s + 4.43·23-s − 7.66·24-s − 13.7·26-s − 2.59·27-s + 4.44·28-s − 0.299·29-s + 5.25·31-s + 4.17·32-s − 5.01·33-s + 3.23·34-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.26·3-s + 1.74·4-s − 2.09·6-s + 0.481·7-s − 1.23·8-s + 0.606·9-s − 0.688·11-s + 2.21·12-s + 1.62·13-s − 0.798·14-s + 0.299·16-s − 0.335·17-s − 1.00·18-s − 0.115·19-s + 0.610·21-s + 1.14·22-s + 0.924·23-s − 1.56·24-s − 2.69·26-s − 0.498·27-s + 0.840·28-s − 0.0557·29-s + 0.944·31-s + 0.737·32-s − 0.872·33-s + 0.555·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254983809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254983809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 0.503T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 0.299T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 5.87T + 71T^{2} \) |
| 73 | \( 1 + 2.22T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.561360493997224864989423978190, −8.626466188797004401732744123682, −8.497389479663231742872224712923, −7.76411160278575215765943917895, −6.93544975623253686037425714244, −5.86603486270152400574192382988, −4.36017078034329611128065166574, −3.09400606009099173355544824371, −2.21366268049736824394615075369, −1.07474610490620465416623833147,
1.07474610490620465416623833147, 2.21366268049736824394615075369, 3.09400606009099173355544824371, 4.36017078034329611128065166574, 5.86603486270152400574192382988, 6.93544975623253686037425714244, 7.76411160278575215765943917895, 8.497389479663231742872224712923, 8.626466188797004401732744123682, 9.561360493997224864989423978190
