| L(s) = 1 | − 2-s + 2.73·3-s + 4-s − 3.46·5-s − 2.73·6-s + 2·7-s − 8-s + 4.46·9-s + 3.46·10-s − 1.26·11-s + 2.73·12-s − 6.19·13-s − 2·14-s − 9.46·15-s + 16-s + 3.46·17-s − 4.46·18-s − 4·19-s − 3.46·20-s + 5.46·21-s + 1.26·22-s − 2.73·24-s + 6.99·25-s + 6.19·26-s + 3.99·27-s + 2·28-s + 2.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s − 1.54·5-s − 1.11·6-s + 0.755·7-s − 0.353·8-s + 1.48·9-s + 1.09·10-s − 0.382·11-s + 0.788·12-s − 1.71·13-s − 0.534·14-s − 2.44·15-s + 0.250·16-s + 0.840·17-s − 1.05·18-s − 0.917·19-s − 0.774·20-s + 1.19·21-s + 0.270·22-s − 0.557·24-s + 1.39·25-s + 1.21·26-s + 0.769·27-s + 0.377·28-s + 0.407·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8540007634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8540007634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−14.88473924442352150783791170523, −14.49695354618813353338842050858, −12.68592736895917291908618555044, −11.63504685108745568464965848447, −10.20567439324370312835760630413, −8.889219258795679303993466098496, −7.81695946794534782740370850693, −7.53960387421692564221676113410, −4.36668582812668839678662110613, −2.69845188150092399463413244841,
2.69845188150092399463413244841, 4.36668582812668839678662110613, 7.53960387421692564221676113410, 7.81695946794534782740370850693, 8.889219258795679303993466098496, 10.20567439324370312835760630413, 11.63504685108745568464965848447, 12.68592736895917291908618555044, 14.49695354618813353338842050858, 14.88473924442352150783791170523
