| L(s) = 1 | − 32·2-s − 827.·3-s + 1.02e3·4-s − 553.·5-s + 2.64e4·6-s + 2.24e4·7-s − 3.27e4·8-s + 5.08e5·9-s + 1.77e4·10-s + 1.61e5·11-s − 8.47e5·12-s − 8.91e4·13-s − 7.17e5·14-s + 4.58e5·15-s + 1.04e6·16-s − 7.88e6·17-s − 1.62e7·18-s + 8.06e6·19-s − 5.67e5·20-s − 1.85e7·21-s − 5.15e6·22-s + 4.63e7·23-s + 2.71e7·24-s − 4.85e7·25-s + 2.85e6·26-s − 2.73e8·27-s + 2.29e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.96·3-s + 0.5·4-s − 0.0792·5-s + 1.39·6-s + 0.503·7-s − 0.353·8-s + 2.86·9-s + 0.0560·10-s + 0.301·11-s − 0.983·12-s − 0.0666·13-s − 0.356·14-s + 0.155·15-s + 0.250·16-s − 1.34·17-s − 2.02·18-s + 0.747·19-s − 0.0396·20-s − 0.991·21-s − 0.213·22-s + 1.50·23-s + 0.695·24-s − 0.993·25-s + 0.0471·26-s − 3.67·27-s + 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 32T \) |
| 11 | \( 1 - 1.61e5T \) |
| good | 3 | \( 1 + 827.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 553.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.24e4T + 1.97e9T^{2} \) |
| 13 | \( 1 + 8.91e4T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.88e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.06e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.63e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.29e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.20e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.89e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.54e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.09e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.53e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.64e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.03e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.70e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.15e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.23e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.59e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.68e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.76e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.34e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.39e10T + 7.15e21T^{2} \) |
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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
−15.32306124853006496141050435896, −12.96539315692386759096857072625, −11.53062442737868598790682331963, −11.04279148988095691392675644732, −9.555487474166650135597638869237, −7.38056513368644883898444628256, −6.12824204512613897784728060392, −4.66690273298168914712788248098, −1.37733475398011802876833281057, 0,
1.37733475398011802876833281057, 4.66690273298168914712788248098, 6.12824204512613897784728060392, 7.38056513368644883898444628256, 9.555487474166650135597638869237, 11.04279148988095691392675644732, 11.53062442737868598790682331963, 12.96539315692386759096857072625, 15.32306124853006496141050435896
