L(s) = 1 | − 3-s − 7-s − 2·9-s + 6·11-s + 4·13-s − 6·17-s + 5·19-s + 21-s − 23-s − 5·25-s + 5·27-s − 8·31-s − 6·33-s + 8·37-s − 4·39-s + 7·41-s + 10·43-s + 5·47-s + 49-s + 6·51-s − 11·53-s − 5·57-s − 2·61-s + 2·63-s + 9·67-s + 69-s − 15·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 25-s + 0.962·27-s − 1.43·31-s − 1.04·33-s + 1.31·37-s − 0.640·39-s + 1.09·41-s + 1.52·43-s + 0.729·47-s + 1/7·49-s + 0.840·51-s − 1.51·53-s − 0.662·57-s − 0.256·61-s + 0.251·63-s + 1.09·67-s + 0.120·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.798713992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.798713992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.49072662590222, −14.07878733017627, −13.62272920134006, −13.08206594738746, −12.39066801401223, −11.99581575586101, −11.29594073734168, −11.15800367633197, −10.75436363315903, −9.665840393103859, −9.329601043030239, −8.993953366962578, −8.378389674443431, −7.599511795469580, −7.052527309999259, −6.357546604297677, −5.985491353560417, −5.716248059710127, −4.694022003355096, −4.080388013744192, −3.685966503025467, −2.907116018830050, −2.060795406203011, −1.246832398091718, −0.5410907872285886,
0.5410907872285886, 1.246832398091718, 2.060795406203011, 2.907116018830050, 3.685966503025467, 4.080388013744192, 4.694022003355096, 5.716248059710127, 5.985491353560417, 6.357546604297677, 7.052527309999259, 7.599511795469580, 8.378389674443431, 8.993953366962578, 9.329601043030239, 9.665840393103859, 10.75436363315903, 11.15800367633197, 11.29594073734168, 11.99581575586101, 12.39066801401223, 13.08206594738746, 13.62272920134006, 14.07878733017627, 14.49072662590222