L(s) = 1 | − 8.20·2-s + 35.3·4-s − 29.2·5-s − 27.7·8-s + 239.·10-s − 377.·11-s + 509.·13-s − 904.·16-s − 1.60e3·17-s + 2.15e3·19-s − 1.03e3·20-s + 3.09e3·22-s + 4.43e3·23-s − 2.27e3·25-s − 4.18e3·26-s − 4.77e3·29-s − 7.15e3·31-s + 8.31e3·32-s + 1.31e4·34-s + 5.57e3·37-s − 1.77e4·38-s + 811.·40-s + 9.91e3·41-s − 4.88e3·43-s − 1.33e4·44-s − 3.64e4·46-s − 9.26e3·47-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.10·4-s − 0.522·5-s − 0.153·8-s + 0.758·10-s − 0.940·11-s + 0.836·13-s − 0.883·16-s − 1.34·17-s + 1.37·19-s − 0.578·20-s + 1.36·22-s + 1.74·23-s − 0.726·25-s − 1.21·26-s − 1.05·29-s − 1.33·31-s + 1.43·32-s + 1.95·34-s + 0.669·37-s − 1.98·38-s + 0.0801·40-s + 0.920·41-s − 0.402·43-s − 1.03·44-s − 2.53·46-s − 0.611·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5610354841\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5610354841\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 8.20T + 32T^{2} \) |
| 5 | \( 1 + 29.2T + 3.12e3T^{2} \) |
| 11 | \( 1 + 377.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 509.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.60e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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
−10.22856499981680309066893297429, −9.198472013588200167456640590183, −8.698092993555542857373564258041, −7.58528305537245538532108075881, −7.19619040945382367681675001810, −5.75348354451856441772174188175, −4.48054948826344086378089647401, −3.07588185089115723203885900992, −1.70618293776182515758503806292, −0.48765561596250828757499328245,
0.48765561596250828757499328245, 1.70618293776182515758503806292, 3.07588185089115723203885900992, 4.48054948826344086378089647401, 5.75348354451856441772174188175, 7.19619040945382367681675001810, 7.58528305537245538532108075881, 8.698092993555542857373564258041, 9.198472013588200167456640590183, 10.22856499981680309066893297429