| L(s) = 1 | − 0.381·2-s − 1.85·4-s − 7-s + 1.47·8-s − 3.47·11-s + 5.23·13-s + 0.381·14-s + 3.14·16-s − 5.70·17-s + 1.23·19-s + 1.32·22-s − 5·23-s − 2·26-s + 1.85·28-s + 8.70·29-s − 4.47·31-s − 4.14·32-s + 2.18·34-s − 3.47·37-s − 0.472·38-s + 8·41-s + 3.76·43-s + 6.43·44-s + 1.90·46-s − 2.76·47-s + 49-s − 9.70·52-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.927·4-s − 0.377·7-s + 0.520·8-s − 1.04·11-s + 1.45·13-s + 0.102·14-s + 0.786·16-s − 1.38·17-s + 0.283·19-s + 0.282·22-s − 1.04·23-s − 0.392·26-s + 0.350·28-s + 1.61·29-s − 0.803·31-s − 0.732·32-s + 0.373·34-s − 0.570·37-s − 0.0765·38-s + 1.24·41-s + 0.573·43-s + 0.970·44-s + 0.281·46-s − 0.403·47-s + 0.142·49-s − 1.34·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9311833326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9311833326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 3.47T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−9.348173658778707124473277094149, −8.548379624885392315692319611720, −8.172581544625168907373866503354, −7.07648225346657981656306154729, −6.11587872576812019024715464593, −5.31400977333011272118104632219, −4.32591314186731984488154729504, −3.57727275723253702872333039579, −2.27657767780759479104994001238, −0.70466716119547569152274873109,
0.70466716119547569152274873109, 2.27657767780759479104994001238, 3.57727275723253702872333039579, 4.32591314186731984488154729504, 5.31400977333011272118104632219, 6.11587872576812019024715464593, 7.07648225346657981656306154729, 8.172581544625168907373866503354, 8.548379624885392315692319611720, 9.348173658778707124473277094149
