| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 27-s − 28-s + 29-s + 30-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 27-s − 28-s + 29-s + 30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4395929735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4395929735\) |
| \(L(1)\) |
\(\approx\) |
\(0.6627353910\) |
| \(L(1)\) |
\(\approx\) |
\(0.6627353910\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−43.988308502403416504944841263, −42.788261511645281673975244898140, −41.95272842208549146545251634782, −39.03678471619023999728500556745, −38.38532547892825026671651055760, −36.727547051537073200591290509603, −35.76319303421611596021350693284, −34.46644233774365394408596282416, −32.4234172849637336609819268055, −31.01874209224480391870730181195, −29.39145603391187839555762592893, −27.64980842472696812713008959723, −26.38431352575075899278988900685, −25.37170440577142621541053358349, −23.59202788517293715038687724139, −20.95918191740503118143937053273, −19.54804144334703986992466920641, −18.75125235623423291645518570775, −16.2748260574985884178954092949, −15.14833241700574422460628551477, −12.61701279102317873036853708840, −10.33642072623153902983172482625, −8.62542663503259159734490395156, −7.23159073941876201502754170862, −3.119341479008603413901599756715,
3.119341479008603413901599756715, 7.23159073941876201502754170862, 8.62542663503259159734490395156, 10.33642072623153902983172482625, 12.61701279102317873036853708840, 15.14833241700574422460628551477, 16.2748260574985884178954092949, 18.75125235623423291645518570775, 19.54804144334703986992466920641, 20.95918191740503118143937053273, 23.59202788517293715038687724139, 25.37170440577142621541053358349, 26.38431352575075899278988900685, 27.64980842472696812713008959723, 29.39145603391187839555762592893, 31.01874209224480391870730181195, 32.4234172849637336609819268055, 34.46644233774365394408596282416, 35.76319303421611596021350693284, 36.727547051537073200591290509603, 38.38532547892825026671651055760, 39.03678471619023999728500556745, 41.95272842208549146545251634782, 42.788261511645281673975244898140, 43.988308502403416504944841263
