L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 2·9-s + 12-s − 13-s − 14-s + 16-s − 2·18-s − 19-s − 21-s + 3·23-s + 24-s − 5·25-s − 26-s − 5·27-s − 28-s + 5·31-s + 32-s − 2·36-s − 4·37-s − 38-s − 39-s − 42-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.471·18-s − 0.229·19-s − 0.218·21-s + 0.625·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + 0.898·31-s + 0.176·32-s − 1/3·36-s − 0.657·37-s − 0.162·38-s − 0.160·39-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733348049\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733348049\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.32248972492364302359117998365, −12.24215793074211449610412193014, −11.31500525007427707811465749128, −10.08859194652768583210503646826, −8.920152760385419156767633970909, −7.79510581397524982652968495375, −6.52013654011009885219318723813, −5.30767778357765891574349430596, −3.77653001984849480071749037204, −2.51874373869269853485475383946,
2.51874373869269853485475383946, 3.77653001984849480071749037204, 5.30767778357765891574349430596, 6.52013654011009885219318723813, 7.79510581397524982652968495375, 8.920152760385419156767633970909, 10.08859194652768583210503646826, 11.31500525007427707811465749128, 12.24215793074211449610412193014, 13.32248972492364302359117998365