| L(s) = 1 | + 2-s + 0.724·3-s + 4-s − 1.67·5-s + 0.724·6-s − 2.46·7-s + 8-s − 2.47·9-s − 1.67·10-s − 1.67·11-s + 0.724·12-s − 2.46·14-s − 1.21·15-s + 16-s + 6.14·17-s − 2.47·18-s − 19-s − 1.67·20-s − 1.78·21-s − 1.67·22-s + 4.29·23-s + 0.724·24-s − 2.19·25-s − 3.96·27-s − 2.46·28-s + 8.69·29-s − 1.21·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.418·3-s + 0.5·4-s − 0.749·5-s + 0.295·6-s − 0.930·7-s + 0.353·8-s − 0.825·9-s − 0.529·10-s − 0.505·11-s + 0.209·12-s − 0.658·14-s − 0.313·15-s + 0.250·16-s + 1.48·17-s − 0.583·18-s − 0.229·19-s − 0.374·20-s − 0.389·21-s − 0.357·22-s + 0.895·23-s + 0.147·24-s − 0.438·25-s − 0.763·27-s − 0.465·28-s + 1.61·29-s − 0.221·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.281782690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281782690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.724T + 3T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 8.49T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 7.59T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 + 6.30T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 - 9.50T + 83T^{2} \) |
| 89 | \( 1 + 5.47T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.001590134323666760138885370426, −7.28321460969623815001006404463, −6.62993703527396558852585257861, −5.73081715202716194476360311389, −5.27752816703660144093813639223, −4.26510835111472774260540042800, −3.30495848542689822459093883787, −3.21236298276294874668836178166, −2.17861336031228783943520727828, −0.66027069550533566538460075984,
0.66027069550533566538460075984, 2.17861336031228783943520727828, 3.21236298276294874668836178166, 3.30495848542689822459093883787, 4.26510835111472774260540042800, 5.27752816703660144093813639223, 5.73081715202716194476360311389, 6.62993703527396558852585257861, 7.28321460969623815001006404463, 8.001590134323666760138885370426
