| L(s) = 1 | − 9·3-s − 81·5-s + 88·7-s + 81·9-s + 399·11-s − 481·13-s + 729·15-s − 289·17-s + 2.56e3·19-s − 792·21-s − 5.01e3·23-s + 3.43e3·25-s − 729·27-s − 378·29-s + 7.60e3·31-s − 3.59e3·33-s − 7.12e3·35-s + 5.75e3·37-s + 4.32e3·39-s − 1.61e4·41-s − 1.26e4·43-s − 6.56e3·45-s + 7.00e3·47-s − 9.06e3·49-s + 2.60e3·51-s + 1.75e4·53-s − 3.23e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.44·5-s + 0.678·7-s + 1/3·9-s + 0.994·11-s − 0.789·13-s + 0.836·15-s − 0.242·17-s + 1.63·19-s − 0.391·21-s − 1.97·23-s + 1.09·25-s − 0.192·27-s − 0.0834·29-s + 1.42·31-s − 0.574·33-s − 0.983·35-s + 0.691·37-s + 0.455·39-s − 1.50·41-s − 1.04·43-s − 0.482·45-s + 0.462·47-s − 0.539·49-s + 0.140·51-s + 0.860·53-s − 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.078136880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078136880\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 17 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 81 T + p^{5} T^{2} \) |
| 7 | \( 1 - 88 T + p^{5} T^{2} \) |
| 11 | \( 1 - 399 T + p^{5} T^{2} \) |
| 13 | \( 1 + 37 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 2569 T + p^{5} T^{2} \) |
| 23 | \( 1 + 5013 T + p^{5} T^{2} \) |
| 29 | \( 1 + 378 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7606 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5756 T + p^{5} T^{2} \) |
| 41 | \( 1 + 16167 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12641 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7002 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17598 T + p^{5} T^{2} \) |
| 59 | \( 1 + 23094 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13876 T + p^{5} T^{2} \) |
| 67 | \( 1 + 42788 T + p^{5} T^{2} \) |
| 71 | \( 1 + 20772 T + p^{5} T^{2} \) |
| 73 | \( 1 + 56626 T + p^{5} T^{2} \) |
| 79 | \( 1 - 85450 T + p^{5} T^{2} \) |
| 83 | \( 1 + 76374 T + p^{5} T^{2} \) |
| 89 | \( 1 - 3468 T + p^{5} T^{2} \) |
| 97 | \( 1 - 48260 T + p^{5} T^{2} \) |
| 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
−9.594608953182645766828960632059, −8.412891372467426289452735622847, −7.74961222734113473399898893383, −7.05968849622141742015539352113, −6.01092063472834923973203508539, −4.82353975688711846522850529683, −4.23292547663686412861416765780, −3.23098027161802988256427239945, −1.63841997976183425823372820511, −0.50171887733892962568841857045,
0.50171887733892962568841857045, 1.63841997976183425823372820511, 3.23098027161802988256427239945, 4.23292547663686412861416765780, 4.82353975688711846522850529683, 6.01092063472834923973203508539, 7.05968849622141742015539352113, 7.74961222734113473399898893383, 8.412891372467426289452735622847, 9.594608953182645766828960632059
