L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s + 11-s − 16-s + 6·17-s + 3·18-s + 4·19-s + 20-s − 22-s + 4·23-s + 25-s + 6·29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s + 2·37-s − 4·38-s − 3·40-s − 2·41-s + 4·43-s − 44-s + 3·45-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s + 0.301·11-s − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s − 0.150·44-s + 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.141206742\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.141206742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82315103384268230837896543123, −7.34274357190881786948937506126, −6.40052201212199996476008390982, −5.60347176654249040201689477012, −4.96297281664656289903473956777, −4.25990473541670011169832908416, −3.30616501936375108866350683234, −2.77081999134239365710804025113, −1.28216188797286466892551129236, −0.67089986211493891609450046840,
0.67089986211493891609450046840, 1.28216188797286466892551129236, 2.77081999134239365710804025113, 3.30616501936375108866350683234, 4.25990473541670011169832908416, 4.96297281664656289903473956777, 5.60347176654249040201689477012, 6.40052201212199996476008390982, 7.34274357190881786948937506126, 7.82315103384268230837896543123