| L(s) = 1 | − 243·3-s + 546.·5-s − 1.68e4·7-s + 5.90e4·9-s + 2.81e5·11-s − 1.34e6·13-s − 1.32e5·15-s + 6.62e6·17-s + 2.91e6·19-s + 4.08e6·21-s − 1.81e6·23-s − 4.85e7·25-s − 1.43e7·27-s + 1.50e8·29-s + 7.40e7·31-s − 6.84e7·33-s − 9.17e6·35-s + 6.30e8·37-s + 3.27e8·39-s − 8.99e8·41-s − 1.77e9·43-s + 3.22e7·45-s + 2.53e9·47-s + 2.82e8·49-s − 1.60e9·51-s − 5.24e9·53-s + 1.53e8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0781·5-s − 0.377·7-s + 0.333·9-s + 0.527·11-s − 1.00·13-s − 0.0451·15-s + 1.13·17-s + 0.269·19-s + 0.218·21-s − 0.0587·23-s − 0.993·25-s − 0.192·27-s + 1.36·29-s + 0.464·31-s − 0.304·33-s − 0.0295·35-s + 1.49·37-s + 0.581·39-s − 1.21·41-s − 1.84·43-s + 0.0260·45-s + 1.61·47-s + 0.142·49-s − 0.653·51-s − 1.72·53-s + 0.0412·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 546.T + 4.88e7T^{2} \) |
| 11 | \( 1 - 2.81e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.34e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.91e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.81e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.50e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.40e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.30e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 8.99e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.77e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.53e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.24e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.45e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.10e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.71e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.07e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.66e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.89e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.80e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 3.23e10T + 7.15e21T^{2} \) |
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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
−11.72477286332435605645273307097, −10.24549911239704361409779697244, −9.561944582069861014345909772006, −7.985542350630076334033761873694, −6.79425510869708194220398014660, −5.69508259340980468066538759644, −4.45978474987146450460063272351, −2.97650373309859604177934281297, −1.34004902561628246889189315199, 0,
1.34004902561628246889189315199, 2.97650373309859604177934281297, 4.45978474987146450460063272351, 5.69508259340980468066538759644, 6.79425510869708194220398014660, 7.985542350630076334033761873694, 9.561944582069861014345909772006, 10.24549911239704361409779697244, 11.72477286332435605645273307097
