| L(s) = 1 | − 0.00233·2-s − 2.04e3·4-s − 1.09e4·5-s + 6.33e4·7-s + 9.55·8-s + 25.5·10-s − 4.20e5·11-s + 8.23e5·13-s − 147.·14-s + 4.19e6·16-s − 2.83e6·17-s + 9.56e6·19-s + 2.24e7·20-s + 981.·22-s + 4.49e7·23-s + 7.12e7·25-s − 1.92e3·26-s − 1.29e8·28-s + 5.52e7·29-s − 2.26e8·31-s − 2.93e4·32-s + 6.60e3·34-s − 6.93e8·35-s + 2.57e7·37-s − 2.23e4·38-s − 1.04e5·40-s − 3.99e8·41-s + ⋯ |
| L(s) = 1 | − 5.15e − 5·2-s − 0.999·4-s − 1.56·5-s + 1.42·7-s + 0.000103·8-s + 8.08e−5·10-s − 0.787·11-s + 0.615·13-s − 7.34e − 5·14-s + 0.999·16-s − 0.483·17-s + 0.885·19-s + 1.56·20-s + 4.05e−5·22-s + 1.45·23-s + 1.45·25-s − 3.17e − 5·26-s − 1.42·28-s + 0.500·29-s − 1.41·31-s − 0.000154·32-s + 2.49e−5·34-s − 2.23·35-s + 0.0610·37-s − 4.56e − 5·38-s − 0.000161·40-s − 0.537·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| good | 2 | \( 1 + 0.00233T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.09e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 6.33e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.20e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 8.23e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.83e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.56e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.49e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.52e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.26e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.57e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.99e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.52e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.50e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.27e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.43e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 6.42e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.39e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 9.59e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.26e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.61e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.07e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.43e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 3.86e10T + 7.15e21T^{2} \) |
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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
−11.45051327545738885640860025670, −10.75966781253455845585079925045, −8.985139718030422336676559783934, −8.149003717097347325637071485181, −7.42344168493856600061818797493, −5.21317243807508380416191530781, −4.46981472813110044934563788761, −3.30825267836872932861393222816, −1.19024789718975159067726160934, 0,
1.19024789718975159067726160934, 3.30825267836872932861393222816, 4.46981472813110044934563788761, 5.21317243807508380416191530781, 7.42344168493856600061818797493, 8.149003717097347325637071485181, 8.985139718030422336676559783934, 10.75966781253455845585079925045, 11.45051327545738885640860025670
