| L(s) = 1 | + 81·3-s − 1.25e3·5-s + 561.·7-s + 6.56e3·9-s + 8.05e4·11-s + 1.19e5·13-s − 1.01e5·15-s − 6.03e5·17-s + 7.22e5·19-s + 4.54e4·21-s + 2.29e5·23-s − 3.87e5·25-s + 5.31e5·27-s + 7.34e6·29-s + 4.26e6·31-s + 6.52e6·33-s − 7.02e5·35-s − 1.01e7·37-s + 9.64e6·39-s − 1.68e7·41-s − 1.63e7·43-s − 8.20e6·45-s + 4.92e7·47-s − 4.00e7·49-s − 4.89e7·51-s − 8.72e7·53-s − 1.00e8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.895·5-s + 0.0883·7-s + 0.333·9-s + 1.65·11-s + 1.15·13-s − 0.516·15-s − 1.75·17-s + 1.27·19-s + 0.0510·21-s + 0.170·23-s − 0.198·25-s + 0.192·27-s + 1.92·29-s + 0.829·31-s + 0.957·33-s − 0.0791·35-s − 0.887·37-s + 0.667·39-s − 0.930·41-s − 0.729·43-s − 0.298·45-s + 1.47·47-s − 0.992·49-s − 1.01·51-s − 1.51·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.011404975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.011404975\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| good | 5 | \( 1 + 1.25e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 561.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.05e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.19e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.03e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.22e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.29e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.34e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.26e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.01e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.63e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.92e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.14e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.21e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.19e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.34e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.54e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.63e8T + 7.60e17T^{2} \) |
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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
−9.611463221712123361161775138406, −8.735439172767916960739657596771, −8.224020699025473559485449457917, −6.95382557414900508886990030406, −6.37774454959532508493874865147, −4.70891310811190047246398217143, −3.91327693507720904487080171929, −3.13174433769476634883076924477, −1.67445832893887215818380395413, −0.74825561801967656757166016029,
0.74825561801967656757166016029, 1.67445832893887215818380395413, 3.13174433769476634883076924477, 3.91327693507720904487080171929, 4.70891310811190047246398217143, 6.37774454959532508493874865147, 6.95382557414900508886990030406, 8.224020699025473559485449457917, 8.735439172767916960739657596771, 9.611463221712123361161775138406
