| L(s) = 1 | + 0.507·2-s + 0.536·3-s − 1.74·4-s + 0.419·5-s + 0.272·6-s + 3.10·7-s − 1.90·8-s − 2.71·9-s + 0.212·10-s − 3.41·11-s − 0.934·12-s − 5.87·13-s + 1.57·14-s + 0.224·15-s + 2.51·16-s + 0.603·17-s − 1.37·18-s − 8.10·19-s − 0.730·20-s + 1.66·21-s − 1.73·22-s − 0.362·23-s − 1.01·24-s − 4.82·25-s − 2.98·26-s − 3.06·27-s − 5.40·28-s + ⋯ |
| L(s) = 1 | + 0.359·2-s + 0.309·3-s − 0.871·4-s + 0.187·5-s + 0.111·6-s + 1.17·7-s − 0.671·8-s − 0.904·9-s + 0.0673·10-s − 1.02·11-s − 0.269·12-s − 1.62·13-s + 0.421·14-s + 0.0580·15-s + 0.629·16-s + 0.146·17-s − 0.324·18-s − 1.85·19-s − 0.163·20-s + 0.363·21-s − 0.369·22-s − 0.0756·23-s − 0.207·24-s − 0.964·25-s − 0.585·26-s − 0.589·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178492592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178492592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 - 0.507T + 2T^{2} \) |
| 3 | \( 1 - 0.536T + 3T^{2} \) |
| 5 | \( 1 - 0.419T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 0.603T + 17T^{2} \) |
| 19 | \( 1 + 8.10T + 19T^{2} \) |
| 23 | \( 1 + 0.362T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 + 0.845T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.24T + 67T^{2} \) |
| 73 | \( 1 - 6.44T + 73T^{2} \) |
| 79 | \( 1 - 5.45T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 - 9.97T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.081134704854588369374711747227, −7.37741042587072836775892618880, −6.19793631860205972576899862009, −5.57861597490279193393825208174, −4.91914376919338848291984244954, −4.50245088677445902139043159097, −3.62143134240734324584199454051, −2.44414584212332956461212407375, −2.20846305527829213870538544610, −0.47162752632438737978026824669,
0.47162752632438737978026824669, 2.20846305527829213870538544610, 2.44414584212332956461212407375, 3.62143134240734324584199454051, 4.50245088677445902139043159097, 4.91914376919338848291984244954, 5.57861597490279193393825208174, 6.19793631860205972576899862009, 7.37741042587072836775892618880, 8.081134704854588369374711747227
