| L(s) = 1 | + 2-s − 3.34·3-s + 4-s − 3.34·6-s + 7-s + 8-s + 8.19·9-s + 11-s − 3.34·12-s − 6.69·13-s + 14-s + 16-s + 7.19·17-s + 8.19·18-s − 1.84·19-s − 3.34·21-s + 22-s − 1.84·23-s − 3.34·24-s − 6.69·26-s − 17.3·27-s + 28-s + 6.84·29-s + 6·31-s + 32-s − 3.34·33-s + 7.19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.93·3-s + 0.5·4-s − 1.36·6-s + 0.377·7-s + 0.353·8-s + 2.73·9-s + 0.301·11-s − 0.965·12-s − 1.85·13-s + 0.267·14-s + 0.250·16-s + 1.74·17-s + 1.93·18-s − 0.424·19-s − 0.730·21-s + 0.213·22-s − 0.385·23-s − 0.682·24-s − 1.31·26-s − 3.34·27-s + 0.188·28-s + 1.27·29-s + 1.07·31-s + 0.176·32-s − 0.582·33-s + 1.23·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.522981283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522981283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 + 0.503T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 0.300T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 1.30T + 83T^{2} \) |
| 89 | \( 1 + 8.69T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−8.137091844767211361921473200467, −7.43512357576680428099496149498, −6.80092111043314453200358290766, −6.07997970850554864962828416212, −5.47181164200978348997470401181, −4.70056129552237685736205339681, −4.45840620048249879380880844758, −3.10682227067054372354790901337, −1.80569546015633953799619776474, −0.72383910057950670534108486937,
0.72383910057950670534108486937, 1.80569546015633953799619776474, 3.10682227067054372354790901337, 4.45840620048249879380880844758, 4.70056129552237685736205339681, 5.47181164200978348997470401181, 6.07997970850554864962828416212, 6.80092111043314453200358290766, 7.43512357576680428099496149498, 8.137091844767211361921473200467
