L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 11-s + 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 2·20-s − 22-s + 4·23-s − 25-s + 2·26-s + 28-s − 2·29-s − 4·31-s + 32-s + 6·34-s + 2·35-s − 2·37-s − 4·38-s + 2·40-s + 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.447·20-s − 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.291260857\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.291260857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.624008951945022350971422712414, −8.817417727367457424810293491861, −7.82491759382859411774440769361, −7.06103937421558916977777144566, −5.88974476808873252053152463118, −5.62472658216482534683012385296, −4.54468764173297491130716278035, −3.52268894932194979183855581758, −2.43951823946059783270435316404, −1.37340800303348386328024038090,
1.37340800303348386328024038090, 2.43951823946059783270435316404, 3.52268894932194979183855581758, 4.54468764173297491130716278035, 5.62472658216482534683012385296, 5.88974476808873252053152463118, 7.06103937421558916977777144566, 7.82491759382859411774440769361, 8.817417727367457424810293491861, 9.624008951945022350971422712414