| L(s) = 1 | − 2.23·2-s + 3.23·3-s + 3.00·4-s − 2·5-s − 7.23·6-s + 7-s − 2.23·8-s + 7.47·9-s + 4.47·10-s − 11-s + 9.70·12-s − 1.23·13-s − 2.23·14-s − 6.47·15-s − 0.999·16-s + 1.23·17-s − 16.7·18-s − 2.47·19-s − 6.00·20-s + 3.23·21-s + 2.23·22-s − 6.47·23-s − 7.23·24-s − 25-s + 2.76·26-s + 14.4·27-s + 3.00·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.86·3-s + 1.50·4-s − 0.894·5-s − 2.95·6-s + 0.377·7-s − 0.790·8-s + 2.49·9-s + 1.41·10-s − 0.301·11-s + 2.80·12-s − 0.342·13-s − 0.597·14-s − 1.67·15-s − 0.249·16-s + 0.299·17-s − 3.93·18-s − 0.567·19-s − 1.34·20-s + 0.706·21-s + 0.476·22-s − 1.34·23-s − 1.47·24-s − 0.200·25-s + 0.542·26-s + 2.78·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7291368077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7291368077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−14.77113383902928186552317166788, −13.64378596260725423988467239504, −12.18782379390857727025883054502, −10.66696977835995280693652080171, −9.646008721917093568598530969159, −8.655694758854538486237566014906, −7.921452813266244493658886433521, −7.29080846889334209401330189578, −3.96714211787804829284362244522, −2.16581206833890556742345562501,
2.16581206833890556742345562501, 3.96714211787804829284362244522, 7.29080846889334209401330189578, 7.921452813266244493658886433521, 8.655694758854538486237566014906, 9.646008721917093568598530969159, 10.66696977835995280693652080171, 12.18782379390857727025883054502, 13.64378596260725423988467239504, 14.77113383902928186552317166788
